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+{"text":"\\section{Introduction} A  covering of a group $G$ is a family $\\{S_i\\}_{i\\in I}$ of subsets of $G$ such that $G=\\bigcup_{i\\in I}\\,S_i$. The famous result of B.H. Neumann states that if $\\{S_i\\}$ is a finite covering of $G$ by cosets of subgroups, then $G$ is actually covered by the cosets $S_i$ corresponding to subgroups of finite index in $G$ \\cite{neumann}. Therefore whenever a group $G$ is covered by finitely many cosets of subgroups it is natural to expect that some structural information about $G$ can be deduced from the properties of the subgroups. In other words, the general question is to what extent properties of the covering subgroups impact the structure of $G$.\n\nIn recent years some ``verbal\" variations of these questions became a subject of research activity. Given a group-word $w=w(x_1,\\dots,x_n)$, we think of it primarily as a function of $n$ variables defined on any given group $G$. We denote by $w(G)$ the verbal subgroup of $G$ generated by the values of $w$. When the set of all $w$-values in a group $G$ is contained in a union of finitely many subgroups (or cosets of subgroups) we wish to know   whether the properties of the covering subgroups have impact on the structure of the verbal subgroup $w(G)$. The present article deals with the situation when $G$ is a profinite group.\n\nIn the context of profinite groups all the usual concepts of group theory are interpreted topologically. In particular, by a subgroup of a profinite group we mean a closed subgroup. A subgroup is said to be generated by a set $S$ if it is topologically generated by $S$. Thus, the verbal subgroup $w(G)$ in a profinite group $G$ is a minimal closed subgroup containing the set of $w$-values. One important tool for dealing with  the ``covering\" problems in profinite groups is the classical Baire's category theorem (cf \\cite[p.\\ 200]{Ke}): If a locally compact Hausdorff space is a union of countably many closed subsets, then at least one of the subsets has non-empty interior. It follows that if a profinite group is covered by countably many cosets of subgroups, then at least one of the subgroups is open. Thus, in the case of profinite groups we can successfully deal with problems on countable coverings rather than just finite ones.\n\nThe reader can consult the articles \\cite{surveyrendiconti,as,DMS1,DMS-revised,DMS-nilpotent, Snilp} \n for results on countable coverings of word-values by \nsubgroups. One of the results obtained in \\cite{DMS-nilpotent} is that if $w$ is a multilinear commutator and $G$ is a profinite group, \nthen $w(G)$ is finite-by-nilpotent if and only if the set of $w$-values in $G$ is covered by countably many finite-by-nilpotent subgroups \n(see Section \\ref{sec:prelim} for the definition of multilinear commutator). It is easy to see that the above result is no longer true \nif the set of $w$-values in $G$ is covered by countably many cosets of finite-by-nilpotent subgroups. This can be exemplified by any \nprofinite group $G$ having $w(G)$ virtually nilpotent but not finite-by-nilpotent. In the present article we study groups in which \nthe set of $w$-values is covered by countably many cosets of $\\mathcal{C}$-subgroups, where $\\C$ is a class of groups closed under \ntaking subgroups, quotients, and such that in any group the product of finitely many normal $\\C$-subgroups is again a $\\C$-subgroup.\n\nOur main result is as follows. \n\n\\begin{theorem}\\label{main} Let $\\C$ be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely \nmany normal \n$\\C$-subgroups is again a $\\C$-subgroup.  Let $w$ be a multilinear commutator word. The verbal subgroup $w(G)$ of a profinite group $G$ is virtually-$\\mathcal{C}$ if and only if the set of $w$-values in $G$ is covered by countably many cosets of $\\mathcal{C}$-subgroups. \n\\end{theorem}\n\nWe note that many natural classes of groups have the properties as the class $\\C$ in the above theorem. For instance, $\\C$ can be the class of nilpotent, pronilpotent, locally nilpotent, or soluble groups. Further examples include torsion groups and groups of finite rank. \nIt is been known for sometime that if $w$ is a multilinear commutator  and\na profinite group $G$ has countably many soluble subgroups whose union contains all $w$-values, \n then $w(G)$ is virtually soluble \\cite[Theorem 7]{adm2012}. If $G$ has countably many torsion subgroups \n (or subgroups of finite rank) whose union contains all $w$-values, then $w(G)$ is torsion (or of finite rank) \\cite{DMS1}. \n Obviously, Theorem \\ref{main} extends these results. Moreover, in the case where $\\C$ is the class of all finite groups, we obtain that \n the set of $w$-values in a profinite group $G$ is countable if and only if $w(G)$ is finite. This was one of the main results in \\cite[Theorem 1.1]{jpaa2}. \n\nA few words about the tools employed in the proof of Theorem \\ref{main}. Rather specific combinatorial techniques for handling multilinear \ncommutator words were developed in \\cite{fernandez-morigi, DMS1,DMS-revised}.  The present article is based on further refinements of those techniques. It seems that any attempt to prove a result of similar nature for words that are not multilinear commutator words would require a different approach.\n\n\\section{Preliminary results}\\label{sec:prelim}  \n \nThroughout, we use the same symbol to denote a group-theoretical property and the class of groups with that property. If $\\C$ is a class of \ngroups, a virtually-$\\C$ group is a group with a normal $\\C$-subgroup of finite index. \nThe class of virtually-$\\C$ groups will be denoted by $\\C\\F$. \n\nLet $\\C$ be a class of groups closed under taking subgroups, quotients, and such that  in any group the product of finitely many \nnormal $\\C$-subgroups is again a  $\\C$-subgroup. For instance $\\C$ is the class of nilpotent, soluble, or finite groups. The next two lemmas are analogues of \nLemma 2.2 of \\cite{DMS-nilpotent} and Lemma 2.6 of \\cite{DMS-revised}, respectively. Therefore we omit their proofs.\n\n \n\\begin{lemma} \\label{normalclosure} In any group a product of finitely many normal $\\CF$-subgroups is again in $\\CF$.\n\\end{lemma}\n\nIf $A$ is a subset of a group $G$, we write $\\langle A \\rangle$ for the subgroup generated by $A$. If $B$ is another subset, \nwe  denote by $A^B$ the set $\\{a^b\\mid a\\in A\\textrm{ and } b\\in B\\}.$\n\n \\begin{lemma}\nLet $L$ be a subgroup of a profinite group $G$ such that the normalizer $N_G(L)$ is open. \n\\begin{enumerate}\n\\item If $L$ is finite, then $\\langle L^G\\rangle$ is finite. \n\\item If $L$ is in $\\C$ and $H$ is a normal open subgroup of $G$ contained in $N_G(L)$, \nthen $\\langle (L\\cap H)^G\\rangle$ is in $\\C$. \n\\end{enumerate}\n\\end{lemma}\n\nThroughout this section $w=w(x_1,\\dots,x_n)$ is a  multilinear commutator.\nMultilinear commutators are words which are obtained by nesting commutators, but using always different variables. More formally, the word $w(x) = x$ in one variable is a multilinear commutator; if $u$ and $v$ are  multilinear commutators involving different variables then the word $w=[u,v]$ is a multilinear commutator, and all multilinear commutators are obtained in this way.\n \nAn important family of  multilinear commutators is formed by so-called derived words $\\delta_k$,  \n  on $2^k$ variables,   defined recursively by\n$$\\delta_0=x_1,\\qquad \\delta_k=[\\delta_{k-1}(x_1,\\ldots,x_{2^{k-1}}),\\delta_{k-1}(x_{2^{k-1}+1},\\ldots,x_{2^k})].$$\n Of course  $\\delta_k(G)=G^{(k)}$ is the $k$-th  term of the derived series of $G$. \n\nWe recall the following well-known result (see for example \\cite[Lemma 4.1]{S2}). \n\\begin{lemma}\\label{lem:delta_k} Let $G$ be a group and let $w$ be a multilinear commutator on $n$ variables. Then each $\\delta_n$-value is a $w$-value.\n\\end{lemma}\n \nIf $A_1,\\dots,A_n$ are subsets of a group $G$, we write\n$$\\X_w(A_1,\\dots,A_n)$$ to denote the set of all $w$-values $w(a_1,\\dots,a_n)$ with $a_i\\in A_i$. Moreover, we write $w(A_1, \\dots , A_n)$ for the subgroup $\\langle\\X_w(A_1,\\dots,A_n)\\rangle$. Note that if every $A_i$ is a normal subgroup of $G$, then $w(A_1, \\dots , A_n)$ is normal in $G$.\n  \nLet $I$ be a subset of $\\{1,\\dots,n\\}$.\n  Suppose that we have a family $A_{i_1}, \\dots , A_{i_s}$ of subsets of $G$ with indices running  over $I$ and another family \n  $B_{l_1}, \\dots , B_{l_t}$ of subsets with indices  running  over $\\{1, \\dots ,n \\} \\setminus I.$ \n We write \n $$w_I(A_i ; B_l)$$ \n for $w(X_1, \\dots , X_n)$, where $X_k=A_k$ if $k \\in I$, and $X_k=B_k$  otherwise. \n On the other hand, whenever $a_i\\in A_i$ for $i\\in I$ and $b_l\\in B_l$ for $l\\in \\{1,\\dots,n\\}\\setminus I$, the symbol \n $w_I(a_i;b_l)$ stands for the element $w(x_1, \\dots , x_n)$, where $x_k=a_k$ if $k \\in I$, and $x_k=b_k$ otherwise.\n\n\\begin{lemma}\\label{zerobis}\nAssume that $G$ is  a group and $A_1,\\dots,A_n,H$ are normal subgroups of $G$.\nLet $a_i\\in A_i$ \n and  $h_i\\in H\\cap A_i$ for every $i=1,\\dots,n$. \n Let   $j \\in \\{1,\\dots,n\\}$ and set  $I=\\{1,\\dots,n\\}\\setminus\\{j\\}$. \n Then there exists  an element \n $$x \\in \\X_w( a_1(H\\cap A_1), \\dots, a_n(H\\cap A_n))$$\n  such that \n$$w(a_1h_1,\\dots,a_nh_n)= x \\cdot  w_I(a_ih_i;h_j).$$        \n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the number of variables $n$ appearing in $w$.\n If $n=1$ then $w(a_1h_1)=a_1h_1$ and the statement  is trivially true.\n \nSo assume that $n\\geq2$ and let $w=[w_1,w_2]$ where $w_1,w_1$ are multilinear commutators in $s$ and $n-s$ variables, respectively. Write\n$$y=w(a_1h_1,\\dots,a_nh_n)=[y_1,y_2]$$ where $y_1=w_1(a_1h_1,\\dots,a_sh_s),$  and $y_2=w_2(a_{s+1}h_{s+1},\\dots,a_nh_n)$.\n\nAssume also that $j>s$. Then by induction \n$y_2=xh$, where \n$x \\in \\X_{w_2} (a_{s+1}(H\\cap A_{s+1}),\\dots,a_n (H\\cap A_n))$ and \n$$h=w_2(a_{s+1}h_{s+1},\\dots,a_{j-1}h_{j-1},h_j,a_{j+1}h_{j+1},\\dots,a_nh_n).$$\nSo\n$$y=[y_1,y_2]=[y_1,xh]=[y_1, h][y_1, x]^h=[y_1,x]^{h [y_1,h]^{-1}} [y_1,h].$$\nSince $\\tilde h=h [y_1,h]^{-1} \\in H$  and  $a_i \\in A_i$, clearly $[a_i,\\tilde h]\\in H\\cap A_i$ and so \n $$(a_i \\tilde h_i)^{\\tilde h}= a_i[a_i,\\tilde h]h_i^{\\tilde h} \\in a_i ( H\\cap A_i),$$ \n  for every $\\tilde h_i \\in H\\cap A_i$ and every $i$. \n As $x  =w_2(a_{s+1}\\tilde h_{s+1},\\dots,a_n\\tilde h_n),$ for some $\\tilde h_{i}\\in H\\cap A_{i},$\n\n it follows that \n \\begin{eqnarray*}\n [y_1,x]^{\\tilde h} &=& w(a_1h_1,\\dots,a_sh_s, a_{s+1} \\tilde h_{s+1},\\dots,a_n \\tilde h_n)^{\\tilde h} \\\\\n &=& w((a_1h_1)^{\\tilde h},\\dots,(a_sh_s)^{\\tilde h}, (a_{s+1} \\tilde h_{s+1})^{\\tilde h},\\dots,(a_n \\tilde h_n)^{\\tilde h})\n\\end{eqnarray*} \n belongs to $ \\X_w( a_1(H\\cap A_1), \\dots, a_n(H\\cap A_n)),$\n as desired. \n\n\nThe case $1\\le j\\le s$ is similar.\nBy induction $y_1=xh$, where \n$$h=w_1(a_{1}h_{1},\\dots,a_{j-1}h_{j-1},h_j,a_{j+1}h_{j+1},\\dots,a_sh_s)$$ and \n$x \\in \\X_{w_1} (a_{1}(H\\cap A_{1}),\\dots,a_s (H\\cap A_s)).$\nSo\n$\ny=[y_1,y_2]=[xh,y_2]=[x,y_2]^h[h,y_2]. $\nNote that $h\\in H$ and $a_i \\in A_i$, therefore, as above, \n$(a_i \\tilde h_i)^{ h} \\in a_i ( A_i\\cap H)$ for every $\\tilde h_i\\in H\\cap A_i$ and every $i$. So $$[x,y_2]^h \\in\\X_w(a_1(H\\cap A_1),\\dots, a_n(H\\cap A_n))$$ and the result follows.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{uno} Let $H,A_1,\\dots,A_n$ be normal subgroups of a group $G$. Let $V$ be a subgroup of $G$ and $g\\in G$. Assume that for some elements $a_i\\in A_i$, the following holds:\n$$\\X_w( a_1(H\\cap A_1),\\dots,a_n(H\\cap A_n)) \n\\subseteq gV.$$\nLet $I$ be a proper subset  of $\\{1,\\dots,n\\}$. \nThen \n$$w_I(a_i (H\\cap A_i);H\\cap A_l )\\le V.$$\n\\end{lemma} \n\\begin{proof}\nThe proof is by  induction on $n-|I|$, \n so first assume that $I=\\{1,\\dots,n \\}\\setminus\\{j\\}$ for some index $j$. \n\n We will write for short $H_i=H\\cap A_i$, for every $i=1, \\dots ,n$. \n \nConsider $ w(g_1,  \\dots , g_n)$, where $g_i \\in a_i  H_i   $ for every $i\\ne j$ and $g_j  \\in H_j$. By Lemma \\ref{zerobis} we have $$w(g_1,  \\dots ,g_{j-1},a_jg_j,g_{j+1},\\dots,g_n)=xw(g_1,\\dots,g_n),$$\nfor some $x\\in \\X_w( a_1  H_1   ,\\dots,a_n  H_n   )\\subseteq gV$. As  \n$$w(g_1,\\dots,a_jg_j,\\dots,g_n)\\in\\X_w(a_1  H_1   ,\\dots,a_n  H_n   ),$$ it follows that $w(g_1,\\dots,g_n)\\in V$. Since $V$ is subgroup, we deduce that $w_I(a_i   H_i   ;H_l )\\le V$ and this concludes the case $|I |=n-1$.  \n\nNow assume that $|I|\\le n-2$ and let $I^*=I\\cup\\{j\\}$ for some $j\\notin I$.\nConsider $w(g_1,\\dots,g_n)$, where $g_i\\in H_i$ for every $i\\in I$ and  $g_i \\in a_i  H_i   $ for every $i\\not\\in I$. Then the element $w(g_1,\\dots,g_{j-1},a_jg_j,g_{j+1},\\dots,g_n)$ belongs to $ w_{I*}(a_i  H_i   ;H_l )$. By Lemma \\ref{zerobis} we have $$w(g_1,\\dots,g_{j-1},a_jg_j,g_{j+1},\\dots,g_n)=xw(g_1,\\dots,g_n),$$  for some \n$$x\\in \\X_w( g_1  H_1   ,\\dots ,g_{j-1}  H_{j-1}   ,a_j  H_j   ,g_{j+1}  H_{j+1}   , \\dots, g_n  H_n   ).$$ In particular $x \\in  w_{I*}(a_i   H_i   ;H_l )$. \nSince, by induction,\n  $ w_{I*}(a_i   H_i   ;H_l )\n\\le V$, it follows that $ w(g_1,\\dots,g_n)\\in V$, as we wanted. The proof is complete.\n\\end{proof} \n\nBy applying the previous lemma with $I=\\emptyset$ and $A_i=G$ for each $i$,   we obtain the following corollary. \n\n\\begin{corollary}\\label{w(H)}\nLet $G$ be a group, $H$ and $V$ subgroups of $G$, and $g\\in G$. Assume that $H$ is normal and $$\\X_w(a_1H,\\dots,a_nH) \\subseteq gV$$ for some elements\n$a_1,\\dots,a_n\\in G$. Then $w(H)\\subseteq V$.\n\\end{corollary}\nThe next lemma is Lemma 4.1 in \\cite{DMS-revised}.\n\n\\begin{lemma}\\label{M} \nLet $A_1,\\dots,A_n$ and $H$ be normal subgroups of  a group $G$. \nLet $I$ be a subset of $\\{1, \\dots ,n \\}$. Assume that for every proper subset $J$ of $I$ \n\\[ w_J (A_i; H\\cap A_l)=1.\\]\nSuppose we are given elements  $g_i \\in A_i$ with $i \\in I$ and  elements $h_k \\in H\\cap A_k$ with $k \\in \\{1, \\dots, n\\}$. \n Then we have \n\\[w_I(g_ih_i; h_l)=w_I(g_i;h_l).\\] \n\\end{lemma}\n\nWe will now introduce some more notation to handle some particular properties of multilinear commutators. We denote by $\\mathbf{I}$ the set of $n$-tuples $(i_1,\\dots,i_n)$, where all entries $i_k$ are non-negative integers. We will view $\\mathbf{I}$ as a partially ordered set with the partial order given by the rule that $$(i_1,\\dots,i_n)\\leq(j_1,\\dots,j_n)$$ if and only if $i_1\\leq j_1,\\dots,i_n\\leq j_n$.\n\nGiven \n $\\mathbf{i}=(i_1,\\ldots,i_n) \\in \\mathbf{I}$, we write\n\\[\nw(\\mathbf{i})=w(G^{(i_1)},\\ldots,G^{(i_n)})\n\\]\n for  the subgroup generated by the $w$-values $w(g_1,\\dots,g_n)$ with  $g_{j} \\in G^{(i_j)}$. \nFurther, let \n\\[\nw(\\mathbf{i^+})=\\prod w(\\mathbf{j} ),\n\\]\n where the product is taken over all $\\mathbf{j} \\in \\mathbf{I} $ such that $\\mathbf{j}>\\mathbf{i}$.\n \n\n\\begin{lemma}\\label{lem:ab} \\cite[Corollary 6]{DMS1}\n Let $w=w(x_1,\\ldots,x_n)$ be a multilinear commutator\\  and let $\\mathbf{i} \\in \\mathbf{I}$. If $ w(\\mathbf{i^+})=1$, then $ w(\\mathbf{i})$ \n is abelian.  \n\\end{lemma}\n\n\nThe following lemma is Proposition 7 in \\cite{DMS1}.\n  \\begin{lemma}\\label{pow_old}\nLet $\\mathbf{i}=(i_1,\\ldots,i_n) \\in \\mathbf{I}$ and suppose that $w(\\mathbf{i^+})=1$. \nIf  $a_j\\in G^{(i_j)}$ for $j = 1,\\dots, n$, and   $b_s\\in  G^{(i_s)}$ \n then \n$$w(a_1 ,\\dots, a_{s-1}, b_s a_s , a_{s+1} ,\\dots, a_k )$$\n$$\\quad=w(\\tilde a_1 ,\\dots, \\tilde a_{s-1}, b_s ,\\tilde  a_{s+1} ,\\dots,\\tilde  a_k )w(a_1 ,\\dots, a_{s-1}, a_s , a_{s+1} ,\\dots, a_k ),$$\nwhere $\\tilde a_j$ is a conjugate of $a_j$ and moreover $\\tilde a_j=a_j$ if $i_j\\le i_s$ .\n \\end{lemma}\n \n\\begin{corollary}\\label{pow}\nAssume that $w(\\mathbf{i^+})=1$ and let $a_j\\in G^{(i_j)} $ for $j=1,\\dots,n$. Let $l$ be an integer. Then $w(a_1,\\dots,a_n)^l=w(b_1,\\dots,b_n)$ for some $b_1,\\dots,b_n$ with $b_j\\in G^{(i_j)}$.\n\\end{corollary}\n\\begin{proof}\n Let $i_s$ be maximal among all $i_j$'s, with $j = 1,\\dots, k$. \n Note that by Lemma \\ref{pow_old} for every  $a_j\\in G^{(i_j)}$, where \n $j=1,\\dots,n$, and every $b_s\\in G^{(i_s)}$ we have:\n $$w(a_1 ,\\dots, a_{s-1}, b_s a_s , a_{s+1} ,\\dots, a_k )$$\n$$\\quad=w(a_1 ,\\dots, a_{s-1}, b_s ,a_{s+1} ,\\dots,a_k )w(a_1 ,\\dots, a_{s-1}, a_s , a_{s+1} ,\\dots, a_k ).$$ It follows that $$w(a_1,\\dots,a_{s-1},  a_s,a_{s+1},\\dots,a_k )^l=w(a_1,\\dots,a_{s-1},a_s^l,a_{s+1},\\dots,a_k )$$ for every integer $l$. This proves the result.\n\\end{proof}\nRecall that an element of a group $G$ is called an $FC$-element if it has only finitely many conjugates in $G$. The next result is Lemma 2.7 in \\cite{DMS-nilpotent}.  \n\n \\begin{lemma}\\label{FC}\nLet $G=\\langle H, a_1,\\dots,a_s\\rangle$ be a profinite group, where $H$ is an open abelian normal \nsubgroup and  $a_1, \\dots, a_s$ are FC-elements. Then $G'$ is finite.\n \\end{lemma}\n\n\n\\section{Proof of the main theorem}\n\nRecall that $\\C$ is a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal $\\C$-subgroups is again a $\\C$-subgroup.\n \nThroughout this section we will work under the following hypothesis:\n\n\\begin{hypothesis}\\label{hyp} Let $w=w(x_1,\\dots,x_n)$ be a multilinear commutator and let $G$ be a profinite group in which the set of $w$-values is contained in a union of countably many cosets $\\g_iG_i$ of\nsubgroups $G_i$, where each $G_i\\in\\C$. \n\\end{hypothesis}\n\n\\begin{lemma}\\label{H} Assume Hypothesis \\ref{hyp}. Then $G$ contains an open normal subgroup $H$ such that $w(H)$ is in $\\C$.\n\\end{lemma}\n\\begin{proof}\nFor each positive integer $i$ consider the set  \\[S_i=\\{ (g_1, \\dots, g_n)\\in G\\times \\dots \\times G \\mid  w(g_1, \\dots, g_n)\\in \\g_iG_i \\}.\\]\nNote that the sets $S_i$ are closed in $G\\times \\dots \\times G$ and cover the whole group $G\\times \\dots \\times G$. By the Baire category theorem at least one of these sets has non-empty interior. Hence, there exist an open \nnormal subgroup $H$ of  $G$, elements $a_1,\\dots a_n \\in G$, and an integer $j$ such that $w(a_1H,\\dots,a_nH)\\subseteq\\g_jG_j$.\nBy  Corollary \\ref{w(H)} we have $w(H)\\le G_j$, so the result follows.\n\\end{proof}\n \n\\begin{lemma}\\label{H_a-multi}  Assume Hypothesis \\ref{hyp} and let $a\\in G$ be a $w$-value. There exists a normal open subgroup $H_a$ in $G$ such that $[H_a,a]$ is in $\\C$.\n\\end{lemma}\n\\begin{proof} For each positive integer $i$ let \n\\[S_i=\\{ x \\in G  \\mid  a^x\\in \\g_iG_i\\}.\\]\nNote that the sets $S_i$ are closed in $G $ and cover the whole group $G$. By the Baire category theorem at least one of these sets has non-empty interior. Hence, there exist an open normal subgroup $H$ \nof $G$, an element $b\\in G$, and an integer $j$  such that $a^{hb}\\in \\g_jG_j$ for any $h\\in H$. Of course we can assume that $\\g_j=a^b$, so that   $a^{-b} a^{hb}\\in G_j$ for every $h\\in H$. Thus $a^{-1}a^h\\in G_j^b$ for every $h\\in H$. Hence, $[a,H]=[H,a]\\le G_j^b$ is in $\\C$. \n\\end{proof}\n\nRecall that $G^{(i)}$ denotes the $i$-th term of the derived series of a group $G$.\n\\begin{proposition}\\label{delta} Assume Hypothesis \\ref{hyp}. Then $G^{(2n)}$ is in $\\C\\F$.\n\\end{proposition}\n\n\\begin{proof} By Lemma \\ref{H} there exists an open normal subgroup $H$ such that $w(H)$ is in $\\C$. Lemma \\ref{lem:delta_k} implies that $H^{(n)}$ is in $\\C$. Let $K=G^{(n)}$ and $L=K\\cap H$. Note that $L$ is open in $K$. Choose a  finite set of $\\delta_n$-values $a_1,\\dots,a_s$ such that\n $K=\\langle L, a_1,\\dots,a_s\\rangle$ and let $H_{a_1},\\dots,H_{a_s}$ be normal open subgroups of $G$ such that $[H_{a_j}, a_j]$ is in $\\C$ for every $j$ (see Lemma \\ref{H_a-multi}). Note that for each $j$ the subgroup $[H_{a_j}, a_j]$ is a normal subgroup of $H_{a_j}$ so $\\langle[H_{a_j},a_j]^G \\rangle$ is in $\\C$. Let $N_1\\le K$ be the subgroup generated by $L^{(n)}$ and the subgroups $\\langle [H_{a_j},a_j]^G\\rangle $ for $j=1,\\dots,s$. Note that $N_1$ is in $\\C$. The images of $a_1,\\dots,a_s$ in the quotient $G\/N_1$ are $FC$-elements while the image of $L$ in $G\/L'$ is abelian. Therefore by \n  Lemma \\ref{FC} the group $KN_1\/L'N_1$ \n has finite \n derived group. In other words $L'N_1$ has finite index in $K'N_1$. \n In particular there exist finitely many  $\\delta_n$-values $b_1,\\dots,b_t$ such that \n   $K'N_1=\\langle L', b_1,\\dots, b_t, N_1\\rangle$.  \n   \n    As above, there exist   normal open subgroups \n $H_{b_1},\\dots,H_{b_t}$ of $G$\n such that $\\langle [H_{b_j},b_j]^G\\rangle $ is in $\\C$ for every $j$.\n  Let $N_2$ be the subgroup generated by $N_1$ \n and the subgroups $\\langle [H_{b_j},b_j]^G\\rangle $ for $j=1,\\dots,t$. Note that\n $N_2$ is in $\\C$. \n  Again, $b_1N_2,\\dots,b_tN_2$ are FC-elements in $G\/N_2$ and arguing as before we\n obtain that $L^{(2)}N_2$ has finite index in $K^{(2)}N_2$. By iterating this argument we get that $L^{(n)}N_n$ has finite index in $K^{(n)}N_n$\n for some normal $\\C$-subgroup $N_n$, so $L^{(n)}(K^{(n)}\\cap  N_n)$ has finite index in $K^{(n)}=G^{(2n)}$. \n As $L^{(n)}\\le H^{(n)}$ is in $\\C$ it follows that $G^{(2n)}$ is in $\\C\\F$, as desired.\n \\end{proof}\n \nRecall the notation introduced in Section \\ref{sec:prelim}: whenever $I$ is a subset\n of $\\{1, \\dots ,n \\}$ and $A_{i_1}, \\dots , A_{i_s}$ and   $B_{l_1}, \\dots , B_{l_t}$  are families \n of subsets of $G$ with indices running  over $I$ and  $\\{1, \\dots ,n \\} \\setminus I$, respectively,  we  write \n  $$w_I(A_i; B_l)$$ \n for the subgroup $w(X_1, \\dots , X_n)$, where $X_k=A_k$ if $k \\in I$, and $X_k=B_k$ otherwise.\nMoreover, whenever $a_i\\in A_i$ for $i\\in I$ and $b_l\\in B_l$ for $l\\in \\{1,\\dots,n\\}\\setminus I$, the symbol \n $w_I(a_i;b_l)$ stands for the element $w(x_1, \\dots , x_n)$, where $x_k=a_k$ if $k \\in I$, and $x_k=b_k$ otherwise.\n \n Furthermore, given \n $\\mathbf{i}=(i_1,\\ldots,i_n) \\in \\mathbf{I}$, we write\n\\[w(\\mathbf{i})=w(G^{(i_1)},\\ldots,G^{(i_n)})\\]\n for  the subgroup generated by the $w$-values $w(g_1,\\dots,g_n)$ with  $g_{j} \\in G^{(i_j)}$ and we set \n$w(\\mathbf{i^+})=\\prod w(\\mathbf{j} )$, \n where the product is taken over all $\\mathbf{j} \\in \\mathbf{I} $ such that $\\mathbf{j}>\\mathbf{i}$.\n \n\n\n\\begin{lemma}\\label{step1}\nAssume Hypothesis \\ref{hyp}.  Let  $A_1,\\dots,A_n$ be normal subgroups of $G$  and let $I$  be a  proper \nsubset of $\\{1, \\dots, n\\}$. \nAssume that there exist  a normal $\\C\\F$-subgroup $T$ of $G$ and an open normal subgroup $H$  such that:  \n\\begin{itemize}\n\\item[(*)] \n$w_J (A_i; H\\cap A_l) \\le T$ for every proper subset $J$ of $I$.\n\\end{itemize}\nThen for any given set of elements $\\{g_i \\}_{i\\in I }$, where $g_i \\in A_i$,  there exist  an open normal subgroup $U$  of $G$, contained in $H$, and a normal $\\C\\F$-subgroup $N$ of $G$, containing $T$, such that \n$$w_I(g_i; U\\cap A_l) \\le N.$$ \n\\end{lemma}\n\\begin{proof}\nConsider the sets \n\\[S_j=\\{(h_1,\\dots,h_{n})\n\\mid h_k\\in H\\cap A_k \\;{\\textrm{and}}\\; w_I (g_i h_i ; h_l)\\in g_jG_j \\}.\\]\nNote that the sets $S_j$ are closed in the group $(H \\cap A_1)\\times \\cdots\\times( H \\cap A_n)$ and cover \nthe whole group. By the Baire  category theorem at least one of these sets has non-empty interior. Hence, there exist\nan integer $r$, open subgroups\n$V_k$ of $H \\cap A_k$, and elements $b_k\\in H \\cap A_k$ for every \n$k=1,\\dots,n$  such that \n$$w_I(g_i b_iv_i; b_lv_l)\\in \\g_rG_r,$$ \n for every $v_i \\in V_i$.  \n Each subgroup $V_k$ is of the form $V_k=U_k\\cap H\\cap A_k$ where $U_k$ is an open subgroup\nof $G$ and we can assume that $U_k$ is normal in $G$.\nLet $U=U_1\\cap\\dots\\cap U_{n}\\cap H$. Note that $U$ is an open normal subgroup of $G$ contained\nin $H$. \nThus \n$$w_I(g_i b_iu_i; b_lu_l)\\in \\g_rG_r,$$\n for every $u_i \\in  U \\cap A_i$.  \n Now we apply  Lemma \\ref{uno} to pass from the cosets $b_l(U \\cap A_l)$ to the subgroups  $U \\cap A_l$, for every $l \\notin I$. \nIt follows from Lemma \\ref{uno}\n  that \n the subgroup  \n$$K=w_I(g_i b_i(U\\cap A_i); U\\cap A_l)$$\n is contained in  $G_r$ and so it is in $\\C\\F$. \nNote that $K \\le U$. Since $U$ has finite index in $G$ and normalizes $K$, by Lemma \\ref{normalclosure}, $\\langle K^G\\rangle$ is in $\\C$. \n\nSet $N= T \\langle K^G\\rangle$ and note that $N\\in\\C$. Using (*) and the fact \n that $T \\le N$ and $b_i(U\\cap A_i)\\subseteq H\\cap A_i$, we can apply Lemma \\ref{M} to the group $G\/N$. Therefore $$w_I(g_i ; U\\cap A_l)N=w_I(g_i b_i (U\\cap A_i); U\\cap A_l)N.$$ Since $w_I(g_i b_i (U\\cap A_i); U\\cap A_l) \\le N$,   we deduce that $$w_I(g_i  ; U\\cap A_l) \\le N,$$ as desired. \n \\end{proof}\n\n\\begin{lemma}\\label{step2} Assume Hypothesis \\ref{hyp}. Let  $A_1,\\dots,A_n$ be normal subgroups of $G$  and let $I$  be a  proper subset of $\\{1, \\dots,n\\}$. Assume that there exist  a normal $\\C\\F$-subgroup $T$ of $G$ and an open normal subgroup $H$  such that:  \n\\begin{itemize}\n\\item[(*)] \n$w_J (A_i; H\\cap A_l) \\le T$ for every proper subset $J$ of $I$.\n\\end{itemize}\nThen there exist  an open normal subgroup $U$  of $G$, contained in $H$, and a normal $\\C\\F$-subgroup $N$ of $G$, containing $T$, such that \n$$w_I( A_i; U\\cap A_l) \\le N.$$ \n\\end{lemma}\n\\begin{proof}\nFor each $i \\in I$ choose a set $R_i$ of coset representatives of $H\\cap A_i$ in $A_i$. Note that all those sets are finite. We apply Lemma \\ref{step1} to each choice of elements $\\bar g =\\{g_i\\}_{i \\in I},$ with  $g_i \\in R_i $: let $U_{\\bar g}$ and $N_{\\bar g}$ be normal subgroups of $G$ such that $w_I(g_i ; U_{\\bar g} \\cap A_l) \\le N_{\\bar g}$. The existence of the subgroups $U_{\\bar g}$ and $N_{\\bar g}$ is guaranteed by Lemma \\ref{step1}. Remark that there are only a finitely many subgroups $U_{\\bar g}$ and $N_{\\bar g}$. Then  $U =\\cap_{\\bar g} U_{\\bar g}$ is a normal open subgroup of $G$ contained in $H$  and $N= \\prod_{\\bar g} N_{\\bar g}$ is a normal \n$\\C\\F$-subgroup  containing $T$, such that \n $$w_I(g_i ; U \\cap A_l)\\le N$$ for every choice of $g_i\\in R_i$. \nNote that, by condition (*) and  Lemma \\ref{M}, \n\\[ w_I(g_i (H \\cap A_i); U \\cap A_l) =  w_I(g_i ; U \\cap A_l) \\le N. \\]\nSince  $A_i = \\cup_{g_i \\in R_i} g_i (H \\cap A_i)$ for every $i \\in I$, we conclude that  \n $$w_I (A_i ; U \\cap A_l )= \\langle \\cup_{\\bar g} w_I(g_i (H \\cap A_i); U \\cap A_l)\\rangle  \n \\le N,$$\n as desired. \n\\end{proof}\n\n\\begin{lemma}\\label{basic-step}\n Assume Hypothesis \\ref{hyp}. \nAssume that there exist an $n$-tuple  $\\mathbf{i} \\in \\mathbf{I}$,  a normal $\\C\\F$-subgroup $T$ of $G$ and an open normal subgroup $H$  such that:  \n\\begin{itemize}\n\\item $w(\\mathbf{i^+}) \\le T$. \n\\item  $w (H) \\le T.$\n\\end{itemize}\n Then  \n$w( \\mathbf{i} )$ is in $\\C\\F$. \n\\end{lemma}\n\\begin{proof} Let $\\mathbf{i}=(i_1,\\ldots,i_n)$. \n We will write for short $$A_j = G^{(i_j)},$$ for every $j= 1,\\dots,n$.\nIt is enough to prove the following statement: \n for every subset $I$ of $\\{1, \\dots , n\\}$, there exist an open normal subgroup $U_I$ of $G$ contained in $H$ and a normal \n $\\C\\F$-subgroup $N_I$ containing $T$ such that \n $w_I( A_i; U_I\\cap A_l) \\le N_I.$\n \nThe proof is by induction on the size $k$ of $I$. If $k=0$, then $I= \\emptyset$ and $$ w_\\emptyset(A_i ; H \\cap A_i)= w(H\\cap A_1, \\dots , H \\cap A_n) \\le w(H)\\le T.$$\n\nSo assume $k>0$. Let $J_1, \\dots J_s$ \n be  all the  proper subsets of $I$. By induction, for each $t=1,\\dots,s$  there exist an open normal subgroup $U_{t}$ of $G$ contained in $H$ and a normal  $\\C\\F$-subgroup $N_{t}$ containing $T$ such that $w_{J_t}( A_i; U_{t}\\cap A_l) \\le N_{t}$. Let $U=\\cap_t U_{t}$ and $N=\\langle N_{t}\\vert t =1, \\dots , s \\rangle$. Then $$w_J( A_i; U \\cap A_l) \\le N$$ for every proper subset $J$ of $I$. \n\nIf $k\\ne n$ we can apply Lemma \\ref{step2} to $I$. We obtain that there exist an open normal subgroup $U_I$ of $G$  contained in $H$ and a normal \n$\\C\\F$-subgroup $N_I$ containing $T$ such that $w_I( A_i; U_I\\cap A_l) \\le N_I$, as desired. \n \n So we are left with the case when $k=n$, and thus, by definition, \n$w( A_1,\\dots, A_n)=w( \\mathbf{i} ).$ \n \nFor each $i \\in I$ choose a set $R_i$ of coset representatives of $ H\\cap A_i$ in $A_i$. Note that all those sets are finite. We pass to the quotient $\\bar G=G\/N$. By Lemma \\ref{M} for each choice of elements $\\bar g_1,\\dots,\\bar g_n$ with  $\\bar g_i \\in \\bar R_i $ and for each $\\bar h_1,\\dots,\\bar h_n\\in \\bar U\\cap \\bar A_i$, we have \n$$w(\\bar g_1\\bar h_1,\\dots,\\bar g_n\\bar h_n)=w(\\bar g_1,\\dots,\\bar g_n).$$\nSo the set $$\\X_w(\\bar A_1, \\dots ,\\bar A_n)$$ is finite.\n\nBy Lemma \\ref{pow} every power of an element in $\\X_w(\\bar A_1, \\dots ,\\bar A_n)$ is again in $\\X_w(\\bar A_1, \\dots ,\\bar A_n)$. So every  element in $\\X_w(\\bar A_1, \\dots ,\\bar A_n)$ has finite order. Therefore  $w(\\bar A_1,\\dots,\\bar A_n)$ is generated by finitely many elements of finite order, and being abelian by Lemma \\ref{lem:ab}, it is actually finite.\nIt follows that $w(A_1,\\dots,A_n)$ is in $\\C\\F$, as desired.\n\\end{proof}\n\n\nWe are now ready to complete the proof of Theorem \\ref{main}. \n\n\\vspace{8pt}\n\\noindent {\\bf Proof of Theorem \\ref{main}} Obviously, if $w(G)$ is in $\\C\\F$ then the set of $w$-values in $G$ is covered by countably many cosets of $\\mathcal{C}$-subgroups. Therefore we only need to show that if the set of $w$-values is covered by countably many $\\C$-subgroups then  $w(G)$ is in $\\C\\F$. \n\nThus, assume that the set of $w$-values in $G$ is covered by countably many $\\C$-subgroups. Proposition \\ref{delta} states that $G^{(2n)}$ is in $\\C\\F$.\n \n Let $H$ be as in Lemma \\ref{H}. Then $w(H)$ is in $\\C$. \n  Let $T=G^{(2n)}w(H)$. Then $T$ is in $\\C\\F$  by Lemma \\ref{normalclosure}. \n   Since\n $G^{(2n)}\\le T$ it follows that $G\/T$ is soluble. \n \n Thus there exist only finitely many \n${\\bf i}\\in \\mathbf{I}$ such that $w({\\bf i})T\/T\\neq1$. \nBy induction on the number of such $n$-tuples ${\\bf i}$, we will prove that every subgroup $w({\\bf i})$ is in $\\C\\F$. \n\nChoose ${\\bf i}=(i_1,\\dots,i_n)\\in \\mathbf{I}$ such that $w({\\bf i})T\/T\\neq1$ while $w({\\bf j})T\/T=1$ whenever ${\\bf i}<{\\bf j}$. \n\n  Now we apply  Lemma \\ref{basic-step} \n and  we obtain that $w({\\bf i})$ is in $\\C\\F$. Let $N=w({\\bf i})T$.\n Then induction on the number of ${\\bf j}\\in \\mathbf{I}$ such that $w({\\bf j})\\not\\le N$ leads us to the conclusion that $w(G)$ is in $\\C\\F$. \\qed\n\\bigskip\n\n\\noindent ACKNOWLEDGMENTS. The problem of studying groups in which $w$-values are covered by countably many cosets of subgroups was suggested to us by J. S. Wilson. We thank him for suggesting the problem. The third author was supported  by  FAPDF and FINATEC.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAntisite is one of the most studied native defect in III--V compounds, for\ninstance, the EL2 center in GaAs.  In a seminal work, Kami\\'nska et\nal.~\\cite{KaminskaPRL1985} established that the EL2 center exhibits a\ntetrahedral symmetry, formed by an isolated As antisite defect (As$_{\\rm\n  Ga}$).  Afterward, several\nexperimental~\\cite{bardelebenPRB1986,kabirajAPL2005} as well as theoretical\nstudies~\\cite{chadiPRL1988,DabrowskPRL1988,ZhangPRB1993,chadiPRB2003,OvehofPRB2005}\nhave been done aiming to clarify the structural and electronic properties of\nEL2 defect in GaAs.  After more than ten year of investigations, currently\nthere is a general agreement that the stable EL2 center is ruled by an\nisolated As$_{\\rm Ga}$ with $T_d$ symmetry, while the metastable EL2$^M$\nstructure is attributed to the As$_{\\rm Ga}$ atom displaced along the $C_{3v}$\naxis.\n\nIts  is  well  known  that  low temperature  growth  of  other  III--V\nsemiconductors,  not  only  GaAs,  allows the  preparation  of  highly\nnonstoichiometric compounds.   Indeed, anion antisite  defects in InP,\nP$_{\\rm  In}$, has been  identified for  the first  time unambiguously\nthrough the  electron paramagnetic resonance technique  by Kennedy and\nWilsey~\\cite{kenedyAPL1984}.   Dreszer  et  al.~\\cite{dreszerPRB1993},\nusing  Hall,  high--pressure  far--infrared absorption  and  optically\ndetected  magnetic   resonance  measurements  (at   low  temperature),\nverified that  InP has two  dominant donor levels associated  with the\nphosphorus antisite defect. Further investigations, based upon {\\it ab\ninitio} total  energy calculations, proposed the  formation of P$_{\\rm\nIn}$  antisites clusters in  InP bulk~\\cite{tmsrhmPRB1999}.   \n\nThe most  interesting property  on this class  of defects is  the fact\nthat      they      also      exhibit     a      clear      metastable\nbehavior~\\cite{chadiPRL1988,mikuckiPRB2000,CaldasPRL1990}.   A neutral\nanion--antisite defect in III--V compounds has a stable fourfold and a\nmetastable    threefold   interstitial   configuration,    where   the\nanion--antisite      is     displaced      along      the     $C_{3v}$\naxis~\\cite{CaldasPRL1990},  similarly  to   EL2  and  EL2$^M$  defects\nobserved in GaAs.  This center can be photo--excited into a metastable\nstate,  and from  which  it returns  to  the ground  state by  thermal\nactivation.  Thus,  it is expected a  persistent photoconductivity for\nP$_{\\rm In}$ at low temperature.\n\nNowadays the electronic properties of several materials can be tailored\nthroughout the manipulation\/control of their size in the atomic scale. Within\nthis new class of (nano)structured materials, the semiconductors nanowires\n($NW$s) are attracting great deal of interest for future applications in\nseveral types of nanodevices.  In particular, InP-nanowires (InP$NW$s) have\nbeen considered as a potential structure for fabrication of sensors, light\nemitting diodes, and field effect\ntransistors~\\cite{jianfangSci2001,xiangfengNat2001}.  Usually the\nvapor--liquid--solid mechanism, with a gold particle seed, has been utilized\nfor the growth of these nanostructures~\\cite{bhuniaAPL2003}.  These materials\nare quasi--one--dimensional with electrons confined perpendicularly to the\n$NW$ growth direction.  They exhibit interesting electronic and optical\nproperties due to quantum confinement effects, viz.: the size dependence of\nInP$NW$ bandgap~\\cite{hengNatMat2003,TSchmidtPRB2005}.  Recent {\\it ab initio}\ncalculations, performed by Li et al.~\\cite{LiPRL2005}, suggested the formation\nof stable $DX$ center in small GaAs quantum--dots,  dot diameter smaller\nthan $\\sim$15~nm.\n\nIt is quite likely the formation of native defects during the $NW$ growth\nprocess.  Therefore, the structural and electronic properties of those defects\nare important issues to be addressed, in order to improve our understanding of\nnative defects in quasi--1D semiconducting $NW$ systems.\n\nIn  this  paper  we  carried  out  an {\\it  ab  initio}  total  energy\ninvestigation  of antisite  defects  in InP$NW$s.   We  find that  the\nformation of EL2--like defects,  antisite dissociation energy, and the\nFranck--Condon (FC)  energy are  ruled by the  $NW$ diameter.   On the\nother  hand, the antisite  formation energies  are insensitive  to the\n$NW$  diameter, being P$_{\\rm  In}$ antisites  the most  likely defect\ncompared with In$_{\\rm P}$.   At the equilibrium geometry, the P$_{\\rm\nIn}$  atom   in  thin  InP$NW$  (diamenter  of   13~\\AA)  exhibits  an\nenergetically  stable  trigonal  symmetry,  followed by  a  metastable\nconfiguration with P$_{\\rm In}$ in an interstitial position, 1.15~\\AA\\\nfrom the $T_d$  site.  In this case, the  P$_{\\rm In}$--P dissociation\nenergy     is    0.33~eV.      Increasing     the    $NW$     diameter\n(13~$\\rightarrow$~18~\\AA),  we  find  a P$_{\\rm  In}$--P  dissociation\nenergy  of  0.53~eV,  where  P$_{\\rm  In}$ lying  on  the  $T_d$  site\nrepresents the  energetically most stable configuration.   For the InP\nbulk  phase,  the P$_{\\rm  In}$--P  dissociation  energy  is equal  to\n0.87~eV.  Similarly to the bulk phase, P$_{\\rm In}$ in InP$NW$ induces\nthe formation of localized states within the energy bandgap.  However,\nfor such a thin $NW$ system, this defect does not exhibit an EL2--like\nbehavior. On  the other hand, increasing the  $NW$ diameter, EL2--like\ndefects are expected to  occur.  Finally, within a constrained density\nfunctional   approach~\\cite{ArtachoPRL2004},   we   map   the   atomic\ndisplacements  along thin  InP$NW$ upon  single excitation  of P$_{\\rm\nIn}$ induced  states, and calculate the  respective relaxation energy,\n``Franck-Condon'' (FC) shift.\n\n\\section{Theoretical Approach}\n\nOur  calculations  were performed  in  the  framework  of the  density\nfunctional theory  (DFT)~\\cite{kohn}, within the  generalized gradient\napproximation  due to  Perdew, Burke,  and  Ernzerhof~\\cite{PBE}.  The\nelectron--ion interaction was  treated by using norm--conserving, {\\it\nab   initio},  fully   separable   pseudopotentials  \\cite{KL}.    The\nKohn--Sham  wave   functions  were   expanded  in  a   combination  of\npseudoatomic numerical orbitals~\\cite{sankey}.   Double zeta basis set\nincluding polarization  functions (DZP)  was employed to  describe the\nvalence  electrons~\\cite{dzp}.    The  self--consistent  total  charge\ndensity  was obtained  by  using the  SIESTA code~\\cite{siesta}.   The\nInP$NW$  was modeled  within  the supercell  approach,  where the  InP\nbilayers  were piled  up along  the [111]  direction  with periodicity\nlength of $2\\sqrt 3 a$ and diameters of 13 and 18~\\AA\\ ($a$ represents\nthe optimized lattice constant  along the [111] direction of InP$NW$).\nThe $NW$ surface dangling bonds were saturated with hydrogen atoms.  A\nmesh cutoff of 170~Ry was  used for the reciprocal--space expansion of\nthe total charge density, and  the Brillouin zone was sampled by using\none special  {\\bf k}  point. We have  verified the convergence  of our\nresults with respect  to the number and choice of  the special {\\bf k}\npoints.  All atoms  of the nanowire were fully  relaxed within a force\nconvergence criterion of 20 meV\/\\AA.\n\n\\begin{figure}[h]\n\\includegraphics[width= 7cm]{side.eps}\n\\caption{Structural models  of thin InP$NW$, diamenter  of 13~\\AA. (a)\nTop view and  (b) side view.  The atomic  geometry around the antisite\ndefects are indicated in (c) P$_{\\rm In}$, and (d) In$_{\\rm P}$.}\n\\label{side}\n\\end{figure}\n\n\\section{Results and Comments}\n\nFigure~\\ref{side}  presents  the  atomic  structure of  thin  InP$NW$,\ndiameter  of  13~\\AA,  growth  along  the [111]  direction,  top  view\n[Fig.~\\ref{side}(a)] and side  view [Fig.~\\ref{side}(b)] (the hydrogen\natoms   are  not  shown).    Due  to   the  1D   quantum  confinement,\nperpendicularly  to  the $NW$  growth  direction,  the  energy gap  of\nInP$NW$    increases   compared   with    the   bulk    InP   (1.0~eV)\n\\cite{TSchmidtPRB2005}.   We find energy  gaps of  2.8 and  2.2~eV for\n$NW$ diameters  of 13  and 18~\\AA, respectively.   It is  important to\ntake into  account those energy gaps are  underestimated, with respect\nto   the   experimental  measurements,   within   the  DFT   approach.\nFigure~\\ref{Bandas2}(a)  presents electronic  band  structure of  thin\nInP$NW$ for  wave vectors parallel  to the [111]  direction ($\\Gamma$L\ndirection). At the $\\Gamma$ point, the highest occupied states exhibit\nan energy split of 0.31 and  0.20~eV ($a_1 + e$), for $NW$ diameter of\n13 and 18~\\AA,  respectively. The valence band maximum  of bulk InP is\ndescribed   by   a   three--fold   degenerated  $t_2$   state.    Such\n($t_2\\rightarrow   a_1  +  e$)   energy  splitting   is  due   to  the\n$T_d\\rightarrow C_{3v}$  symmetry lowering of InP$NW$  with respect to\nthe bulk phase.\n\n\\begin{figure}[h]\n\\includegraphics[width= 8cm]{Bandas2.eps}\n\\caption{Electronic band  structures of (a) perfect  thin InP$NW$, and\n(b)  defective thin  InP$NW$ with  a P  antisite (P$_{\\rm  In}$). $NW$\ndiameter of 13~\\AA.}\n\\label{Bandas2}\n\\end{figure}\n\nThe energetic  stability of antisites  in InP$NW$s can be  examined by\nthe  calculation of  formation energies  ($\\Omega_i$).   The formation\nenergy of  P and In  antisites, P$_{\\rm In}$  [Fig.~\\ref{side}(c)] and\nIn$_{\\rm P}$ [Fig.~\\ref{side}(d)], respectively, can be written as,\n$$ \\Omega_i =  E[{\\rm InP}NW_i] - E[{\\rm InP}NW]  - n_{\\rm In}\\mu_{\\rm\nIn} - n_{\\rm P}\\mu_{\\rm P}.\n$$\n\nWe have considered the formation of antisites inner InP$NW$s.  $E[{\\rm\nInP}NW_i]$ represents the  total energy of InP$NW$ with  $i$ = P$_{\\rm\nIn}$  or In$_{\\rm  P}$ antisite  defect, and  $E[{\\rm InP}NW]$  is the\ntotal energy of a perfect InP$NW$.  $n_{\\rm In}$ ($n_{\\rm P}$) denotes\nthe number of  In (P) atoms in  excess or in deficiency. The  In and P\nchemical potentials,  $\\mu_{\\rm In}$ and  $\\mu_{\\rm P}$, respectively,\nare  constrained  by  following thermodynamic  equilibrium  condition,\n$\\mu_{\\rm In}  + \\mu_{\\rm P} = \\mu_{\\rm  InP}^{Bulk}$, where $\\mu_{\\rm\nInP}^{Bulk}$ is the chemical potential  of bulk InP.  Under In rich (P\npoor)  condition  we   will  have  $\\mu_{\\rm  In}\\rightarrow  \\mu_{\\rm\nIn}^{Bulk}$,  whereas  under In  poor  (P  rich) condition,  $\\mu_{\\rm\nIn}\\rightarrow     \\mu_{\\rm    In}^{Bulk}     -     \\Delta    H_f({\\rm\nInP})$~\\cite{qianPRB1988}.   For the  heat of  formation of  bulk InP,\n$\\Delta  H_f(\\rm InP)$,  we  have considered  its experimental  value,\n0.92~eV.\n\nFigure~\\ref{FormEnergy} presents our  calculated results of $\\Omega_i$\nfor  thin  InP$NW$ (diameter  of  13~\\AA), as  a  function  of the  In\nchemical potential.  It is clear that the formation of P$_{\\rm In}$ is\ndominant compared  with In$_{\\rm P}$.  This latter  defect occurs only\nfor  In rich  condition.  At  the  In and  P stoichiometric  condition\n(dashed  line  in   Fig.~\\ref{FormEnergy})  we  obtained  $\\Omega_{\\rm\nP_{In}}$ = 2.15~eV and  $\\Omega_{\\rm In_P}$ = 3.57~eV.  Increasing the\n$NW$  diameter  (18~\\AA), we  find  $\\Omega_{\\rm  P_{In}}$ =  2.18~eV.\nWithin the  same calculation procedure, we  obtained similar formation\nenergy results for  the InP bulk phase, viz.:  $\\Omega_{\\rm P_{In}}$ =\n2.14~eV and  $\\Omega_{\\rm In_P}$  = 3.59~eV.  Those  results indicates\nthat, (i) there is an  energetic preference of P$_{\\rm In}$ antisites,\ncompared with In$_{\\rm  P}$, for both structural phases  of InP.  (ii)\nThe  formation energy  of $\\rm  P_{In}$ does  not depend  on  the $NW$\ndiameter.\n\n\n\\begin{figure}[h]\n\\includegraphics[width= 7cm]{FormEnergy.eps}\n\\caption{Formation  energies   of  P$_{\\rm  In}$   and  In$_{\\rm  P}$\nantisites in  thin InP$NW$,  diameter of 13~\\AA,  as a function  of In\nchemical potential.}\n\\label{FormEnergy}\n\\end{figure}\n\nAt the equilibrium geometry, P$_{\\rm In}$ defect in bulk InP keeps the\n$T_d$ symmetry  with P$_{\\rm In}$--P  bond lengths of  2.49~\\AA, while\nIn$_{\\rm P}$  exhibits a weak Jahn--Teller distortion  along the [001]\ndirection.   Those results  are in  accordance with  previous  {\\it ab\ninitio}        studies        of        antisites        in        InP\n\\cite{seitsonenPRB1994,tmsrhmPRB1999,castletonPRB2004}.   On the other\nhand, the equilibrium  geometry of antisites in thin  InP$NW$ is quite\ndifferent.  The P$_{\\rm In}$--P$_1$  bond, parallel to the $NW$ growth\ndirection, is stretched by 27\\%  compared with the other three P$_{\\rm\nIn}$--P$_i$ bonds,  $i=2-4$ in Fig.~\\ref{side}(c).   Similarly for the\nIn$_{\\rm  P}$ antisites,  In$_{\\rm P}$--In$_1$  is stretched  by 8.6\\%\ncompared  with the  other three  In$_{\\rm P}$--In  bonds  indicated in\nFig.~\\ref{side}(d).           Figures~\\ref{rhoTotParc2}(a)         and\n\\ref{rhoTotParc2}(b)  depict  the  total  charge densities  along  the\nP$_{\\rm In}$--P$_1$  and In$_{\\rm P}$--In$_1$  bonds, respectively. In\nparticular,  due to  the  large P$_{\\rm  In}$--P$_1$ bond  stretching,\nP$_{\\rm  In}$ is  weakly bonded  to P$_1$,  whereas the  other P$_{\\rm\nIn}$--P$_2$, --P$_3$ and --P$_4$ bonds [see Fig.~\\ref{rhoTotParc2}(a)]\nexhibit a  strong covalent character.  Therefore,  different from bulk\nInP,  P$_{\\rm  In}$  antisites  in  $NW$ system  exhibits  a  $C_{3v}$\nsymmetry,  with the  P$_{\\rm In}$  atom displaced  from $T_d$  site by\n0.15~\\AA\\  along the [111]  axis.  Such  a P$_{\\rm  In}$ displacement,\nalong the $C_{3v}$ axis, has  not been observed by increasing the $NW$\ndiameter to  18~\\AA.  In this case,  the P$_{\\rm In}$  atom accupies a\n$T_d$ site and the P$_{\\rm In}$--P bond lengths are equal to 2.94~\\AA.\n\n\\begin{figure}[h]\n\\includegraphics[width= 8cm]{TotHomoLumo.eps}\n\\caption{Total charge densities of (a) P$_{\\rm In}$, and (b) In$_{\\rm P}$\n  antisites in InP$NW$. The partial charge densities of P$_{\\rm In}$ induced\n  (c) highest occupied $v0$ and (d) lowest unoccupied $c1$ states.}\n\\label{rhoTotParc2}\n\\end{figure}\n\nSeveral  anion antisite defects  in III--V  materials were  studied by\nCaldas et al.~\\cite{CaldasPRL1990}.  In  that work, based upon {\\it ab\ninitio}   total   energy  calculations,   the   authors  observed   an\nenergetically stable $T_d$ symmetry  for P$_{\\rm In}$ and a metastable\n$C_{3v}$   configuration   for   the   antisite  atom   displaced   by\n$\\sim$1.3~\\AA\\    along    the    [111]   direction.     Indeed,    in\nFig.~\\ref{DisplEnergy2}(a) we  present our  total energy results  as a\nfunction of P$_{\\rm In}$  displacement, along the [111] direction, for\nthe InP  bulk phase.  We find  that the $T_d$  symmetry represents the\nenergetically most stable configuration, followed by an energy barrier\nof 0.87~eV at 0.7~\\AA\\ from the  $T_d$ site, z = 0.7~\\AA, breaking the\nP$_{\\rm   In}$--P$_1$   bond.    Finally,  the   metastable   $C_{3v}$\nconfiguration occurs for z =  1.2~\\AA, where the P$_{\\rm In}$ occupies\nan     interstitial    site.      Figures~\\ref{DisplEnergy2}(b)    and\n\\ref{DisplEnergy2}(c) present  the energy barrier for  P$_{\\rm In}$ in\nthin InP$NW$.   We have examined  two different processes: (i)  the In\nand P  atoms of the $NW$ are  not allowed to relax  during the P$_{\\rm\nIn}$  displacement  [Fig.~\\ref{DisplEnergy2}(b)]. (ii)  The  In and  P\ncoordinates are  fully relaxed  for each P$_{\\rm  In}$ step  along the\n[111]       direction,      i.e.       an       adiabatic      process\n[Fig.~\\ref{DisplEnergy2}(c)].   It  is   noticeable  that  the  energy\nbarrier calculated in (i) is very similar to that obtained for P$_{\\rm\nIn}$   in   bulk  InP   [Fig.~\\ref{DisplEnergy2}(a)].    In  (i)   the\nenergetically most  stable configuration  for P$_{\\rm In}$  exhibits a\n$T_d$ symmetry, and  we find a dissociation energy of  0.84~eV for z =\n0.86~\\AA.   Further  P$_{\\rm In}$  displacement  indicates a  $C_{3v}$\nmetastable geometry  for z $\\approx$ 1.15~\\AA, where  the P$_{\\rm In}$\natom occupies  an interstitial site.  Meanwhile, in  (ii) the $C_{3v}$\nsymmetry, with the P$_{\\rm In}$ atom lying at 0.15~\\AA\\ from the $T_d$\nsite,  represents the  energetically most  stable  configuration.  The\nP$_{\\rm  In}$--P$_1$ dissociation energy  reduces to  0.33~eV (at  z =\n0.85~\\AA),  and  there  is  a  metastable  geometry  for  z  $\\approx$\n1.15~\\AA.  The  total energy difference  between the stable  ($S$) and\nthe  metastable ($M$)  configurations, $E(S)  - E(M)$,  in  InP$NW$ is\nequal   to  $-$0.084~eV,   while  in   bulk  InP   we  find   $E(S)  -\nE(M)$~=~$-$0.53~eV.  Comparing (i) and  (ii) we verify that the atomic\nrelaxation plays a fundamental rule not only to the energy barrier for\nthe P$_{\\rm  In}$ atomic displacement  along the [111]  direction, but\nalso  for the  equilibrium  geometry of  the  P$_{\\rm In}$  structure.\nIncreasing  the $NW$  diameter, the  energy barrier  for  P$_{\\rm In}$\ndisplacement will  become similar  to that obtained  for bulk  InP (an\nupper     limit    for    very     large    $NW$     diameter),    see\nFig.~\\ref{DisplEnergy2}(d).  Indeed,  for $NW$ diameter  of 18~\\AA, we\nfind a P$_{\\rm In}$--P$_1$ dissociation energy of 0.53~eV, and $E(S) -\nE(M)$ equal  to $-$0.14~eV.  Those  results indicate that  the P$_{\\rm\nIn}$--P$_1$ dissociation  barrier, and the $E(S) -  E(M)$ total energy\ndifference,  can  be  tuned  within  the shaded  region  indicated  in\nFig.~\\ref{DisplEnergy2}(d).\n\nFocusing on  the electronic structure,  the formation of  P$_{\\rm In}$\ndefect  in thin  InP$NW$  gives rise  to  a deep  double donor  state,\nlabeled  $v0$ in  Fig.~\\ref{Bandas2}(b), lying  at 1.3~eV  above $v1$.\nSuch  a  P$_{\\rm In}$  induced  state  is  very localized  within  the\nfundamental band gap, being almost flat along the $\\Gamma$L direction.\nFor P$_{\\rm  In}$ in  bulk InP, {\\it  ab initio} studies  performed by\nSeitsonen et al.~\\cite{seitsonenPRB1994} indicates the formation of an\noccupied state at 0.7~eV above  the valence band maximum.  The partial\ncharge density contour  plot of $v0$, Fig.~\\ref{rhoTotParc2}(c), shows\nan  anti-bonding  $\\pi^\\ast$ orbital  concentrated  along the  P$_{\\rm\nIn}$--P$_1$  bond.   The  lowest  unoccupied  state,   $c1$,  is  also\nconcentrated   on  the   P$_{\\rm   In}$  antisite,   as  depicted   in\nFig.~\\ref{rhoTotParc2}(d).\n\n\\begin{figure}[h]\n  \\includegraphics[width= 7cm]{DisplEnergy.eps}\n\\caption{Total Energy as a function of the P$_{\\rm In}$ antisite \n  displacement along the [111] direction: (a) bulk InP, and (b) InP$NW$ where\n  the In and P atoms are not allowed to relax during the P$_{\\rm In}$\n  displacement. (c) Whole In and P atomic position along the $NW$ are allowed\n  to relax. In (c) the total energy barrier indicated by triangles (dashed\n  line) was calculated for an excited electronic configuration, single\n  excitation. (d) Energy barriers for bulk InP and thin InP$NW$ systems.}\n\\label{DisplEnergy2}\n\\end{figure}\n\nEL2 and EL2$^M$ centers in III--V are characterized by a stable $T_d$ and a\nmetastable $C_{3v}$ geometries for an isolated V$_{\\rm III}$ interstitial\natom.  However, before the V$_{\\rm III}$ antisite arrives to the metastable\n$C_{3v}$ configuration, there is a strong electronic coupling between the\nhighest occupied and the lowest unoccupied antisite induced states near to the\nlocal maximum for the V$_{\\rm III}$--V dissociation energy, see Fig.~3 in\nRef.~\\cite{CaldasPRL1990} and Fig.~4 in Ref.~\\cite{DabrowskPRB1989}.\n\nSimilarly,  for thin  InP$NW$,  we examined  the  energy positions  of\nsingle particle  eigenvalues, $\\epsilon(v0)$ and  $\\epsilon(c1)$, as a\nfunction of the P$_{\\rm In}$ displacement along the $C_{3v}$ axis. Our\ncalculated  results, depicted  in Fig.~\\ref{eigenvalues},  reveal that\nthere is  no electronic  coupling between $v0$  and $c1$.  Thus, based\nupon the  calculated energy barrier  [Fig.~\\ref{DisplEnergy2}(c)], and\nthe evolution  of the antisite induced  states, $v0$ and  $c1$, we can\nstate that there is no EL2--like center in thin InP$NW$s. On the other\nhand,  increasing   the  $NW$  diameter  [Fig.~\\ref{DisplEnergy2}(d)],\nEL2--like center is expected to occur in InP.\n\n\\begin{figure}[h]\n\\includegraphics[width= 7cm]{DisplEigenvNew2.eps}\n\\caption{Single particle eigenvalues ($\\epsilon$) for P$_{\\rm In}$: \n  highest occupied $v0$ [Fig.~\\ref{rhoTotParc2}(c)] and lowest unoccupied $c1$\n  [Fig.~\\ref{rhoTotParc2}(d)].}\n\\label{eigenvalues}\n\\end{figure}\n\nWithin the  DFT approach, we have examined  the structural relaxations\nupon single excitation  from the highest occupied state  ($v0$) to the\nlowest unoccupied  state ($c1$) of  P$_{\\rm In}$ in thin  InP$NW$.  We\nhave used the calculation procedure  proposed by Artacho et al., where\nthe single excitation was modeled  by ``promoting an electron from the\nhighest  occupied molecular  orbital (HOMO)  to the  lowest unoccupied\nmolecular     orbital    (LUMO)''\\cite{ArtachoPRL2004},     i.e.     a\n``constrained'' DFT calculation. So  that, promoting one electron from\n$v0$ to $c1$, $v0\\rightarrow c1$, we obtained the equilibrium geometry\n(full relaxations)  as well as the  self--consistent electronic charge\ndensity.    The   atomic   relaxations   along  thin   InP$NW$,   upon\n$v0\\rightarrow  c1$  single  excitation,  are localized  near  to  the\nP$_{\\rm In}$  position.  Since we  are removing one electron  from the\nanti-bonding $\\pi^\\ast$ orbital depicted in Fig.~\\ref{rhoTotParc2}(c),\nthe P$_{\\rm In}$--P$_1$ bond  shrinks by 0.12~\\AA\\ (2.90 $\\rightarrow$\n2.78~\\AA), and there is radial  a contraction of $\\sim$0.2~\\AA\\ at z =\n6.8~\\AA.\n\nThe  structural relaxations  are  proportional to  the degradation  of\noptical energy due to the  atomic displacements associated with such a\n$v0 \\rightarrow c1$ single excitation, FC shift.  Comparing the radial\nand the longitudinal displacements, we  verify that the most of the FC\nshift comes  from the  atomic relaxation parallel  to the  $NW$ growth\ndirection.   The  atomic  displacements  along  the  radial  direction\n($\\rm\\Delta  r$) are localized  nearby P$_{\\rm  In}$, lying  within an\ninterval  of $-0.2<\\Delta  r<0$~\\AA.  While  the  atomic displacements\nparallel to the growth direction ($\\rm\\Delta z$) lie within a range of\n$-0.2<\\Delta  z<0.3~$\\AA, being less  concentrated around  the P$_{\\rm\nIn}$ antisite position.  We find a FC energy shift of 0.8~eV, which is\nquite  large  compared with  the  InP  bulk  phase.  Within  the  same\ncalculation approach,  we find  a FC energy  shift of 0.05~eV  for InP\nbulk.  Previous {\\it  ab initio} study indicates a  FC energy shift of\n0.1~eV for bulk  InP.  Thus, suggesting that the  FC energy shift also\ncan be tuned by controlling the $NW$ diameter.\n\n\\section{Conclusions}\n\nIn  summary,  we  have  performed  an {\\it  ab  initio}  total  energy\ninvestigation of antisites  in InP$NW$. We find that  the P$_{\\rm In}$\nantisite is the  most likely to occur.  For  thin InP$NW$s the P$_{\\rm\nIn}$ atom  exhibits a  trigonal symmetry, followed  by a  metastable P\ninterstitial  configuration.  Increasing  the $NW$  diameter (18~\\AA),\nP$_{\\rm In}$  occupying the $T_d$ site becomes  the energetically most\nstable configuration.  The calculated energy barrier, for P$_{\\rm In}$\ndisplacement  along the  $C_{3v}$  axis, indicates  that  there is  no\nEL2--like  defect  in thin  InP$NW$s.   However,  increasing the  $NW$\ndiameter,  13  $\\rightarrow$  18~\\AA,  we  observe  the  formation  of\nEL2--like defects  in InP.  Within a ``constrained''  DFT approach, we\ncalculated  the structural relaxations  and the  FC energy  shift upon\nsingle excitation  of the electronic  states induced by  P$_{\\rm In}$.\nWe inferred that not only the formation (or not) of EL2--like defects,\nbut also (i)  the P$_{\\rm In}$--P dissociation energy,  (ii) the total\nenergy   difference  between  stable   and  metastable   P$_{\\rm  In}$\nconfigurations,  $E(S)-E(M)$, and  the (iii)  FC energy  shift  can be\ntuned by controlling the InP$NW$ diameter.\n\n\\begin{center}\n  {\\large\\bf Acknowledgments}\n\\end{center}\n\nThe authors acknowledge financial support from the Brazilian agencies CNPq,\nFAPEMIG, and FAPESP. The most of calculations were performed using the\ncomputational facilities of the Centro Nacional de Processamento de Alto\nDesempenho\/CENAPAD-Campinas.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent years, the Discontinuous Galerkin (DG) method has become a popular\nchoice for the discretization of a wide range of partial differential equations\n\\cite{Reed_Hill,cockburn01rkdg,hesthaven08dgbook}. This is partly because of its\nmany attractive properties, such as the arbitrarily high degrees of\napproximation, the rigorous theoretical foundation, and the ability to use fully\nunstructured meshes. Also, due to its natural stabilization mechanism based on\napproximate Riemann solvers, it has in particular become widely used in fluid\ndynamics applications where the high-order accuracy is believed to produce\nimproved accuracy for many problems \\cite{Wang2013}.\n\nMost work on DG methods has been based on meshes of either simplex elements\n(triangles and tetrahedra), block elements (quadrilaterals and hexahedra), or\ncombinations of these such as prism elements. This is likely because of the\navailability of excellent automatic unstructured mesh generators, at least for\nthe simplex case\n\\cite{peraire87advancingfront,ruppert95delaunay,shewchuk02delaunay}, and also\nbecause of the advantages with the outer-product structure of block elements.\nHowever, it is well known that since no continuity is enforced between the\nelements, it is straightforward to apply the DG methods to meshes with elements\nof any shapes (even non-conforming ones). For example, vertex-centered DG\nmethods based on the polygonal dual meshes were studied in\n\\cite{berggren06vertex,luo08taylor}. This is a major advantage over standard\ncontinuous FEM methods, which need significant developments for the extension to\narbitrary polygonal and polyhedral elements \\cite{manzini2014polygonal}.\n\nIn the finite volume CFD community, there has recently been considerable\ninterest in meshes of arbitrary polygonal and polyhedral elements. In fact, the\npopular vertex-centered finite volume method applied to a tetrahedral mesh can\nbe seen as a cell-centered method on the dual polyhedral mesh. Because of this,\na number of methods have been proposed for generation of polyhedral meshes,\nwhich in many cases have advantages over traditional simplex meshes\n\\cite{oaks2000polyhedral,garimella2014polyhedral}. Although it is still unclear\nexactly what benefits these elements provide, they have been reported to be both\nmore accurate per degree of freedom and to have better convergence properties in\nthe numerical solvers than for a corresponding tetrahedral mesh\n\\cite{peric2004polyhedral,balafas2014polyhedral}. There have also been studies\nshowing that vertex-centered schemes are preferred over cell-centered\n\\cite{diskin2010viscous,diskin2011inviscid}, again indicating the benefits of\npolyhedral elements.\n\nInspired by the promising results for polyhedral finite volume method, and the\nfact that DG is a natural higher-order extension of these schemes, in this work\nwe study some of the properties of DG discretizations on polygonal meshes. To\nlimit the scope, we only investigate the convergence properties of iterative\nsolvers for the discrete systems, assuming an equal number of degrees of freedom\nper unit area for all element shapes. Future work will also investigate the\naccuracy of the solutions on the different meshes. We first consider the\niterative block-Jacobi method applied to a pure convection problem, which in the\nconstant coefficient case can be solved analytically using von Neumann analysis.\nNext we apply the solver to Euler's equations of gas dynamics for relevant model\nflow problems, to obtain numerical results for the convergence of the various\nelement shapes. We consider regular meshes of hexagons, squares, and two\ndifferent configurations of triangles, as well as the dual of fully unstructured\ntriangular Delaunay refinement meshes. We also perform numerical experiments\nwith the GMRES Krylov subspace solver and a block-ILU preconditioner. Although\nthe results are not entirely conclusive, most of the results indicate a clear\nbenefit with the hexagonal and quadrilateral elements over the triangular ones.\n\nThe paper is organized as follows. In Section 2, we describe the spatial and the\ntemporal discretizations, and introduce the iterative solvers. In Section 3 we\nperform the von Neumann analysis of the constant coefficient advection problem,\nin 1D and for several mesh configurations in 2D. In Section 4 we show numerical\nresults for more general advection fields, for more general meshes, as well as\nfor the Euler equations and the GMRES solver. We conclude with a summary of our\nfindings as well as directions for future work.\n\n\\section{Numerical methods}\n\\subsection{The discontinuous Galerkin formulation}\n\nWe consider a system of $m$ hyperbolic conservation laws given by the equation\n\\begin{equation}\n    \\label{eq:cons-law}\n    \\begin{cases}\n    \\partial_t \\bm{u} + \\nabla \\cdot \\bm{F}(\\bm{u}) = 0,\n        \\qquad(t,\\bm{x}) \\in [0, T] \\times \\Omega\\\\\n    \\bm{u}(0, \\bm{x}) = \\bm{u}_0(\\bm{x}).\n    \\end{cases}\n\\end{equation}\nIn order to describe the discontinuous Galerkin spatial discretization, we\ndivide the spatial domain $\\Omega \\subseteq \\mathbb{R}^2$ into a collection of\nelements, to form the \\emph{triangulation} $\\mathcal{T}_h = \\{ K_i \\}$. Often the\nelements $K_i$ are considered to be triangles or quadrilaterals, but in this paper\nwe allow the elements to be arbitrary polygons in order to study the impact\nof different tessellations on the efficiency of the algorithm.\n\nLet $V_h = \\left\\{ v_h \\in L_2(\\Omega) : v_h \\big|_{K_i} \\in P^p(K_i) \\right\\}$ \ndenote the space of piecewise polynomials of degree $p$. We let $\\bm{V}_h^m$ \ndenote the space of vector-valued functions of length $m$, with each component \nin $V_h$. Note that continuity is not enforced between the elements.\nWe derive the discontinuous Galerkin method by replacing $\\bm{u}$ in equation \n\\eqref{eq:cons-law} by an approximate solution $\\bm{u}_h \\in \\bm{V}_h^m$, and then \nmultiplying equation  by a test function $\\bm{v}_h \\in \\bm{V}_h^m$. We then \nintegrate by parts over each element. Because the approximate solution $\\bm{u}_h$ \nis potentially discontinuous at the boundary of an element, the flux function \n$\\bm{F}$ is approximated by a \\emph{numerical flux function} $\\widehat{\\bm{F}}$, \nwhich takes as arguments $\\bm{u}^+$, $\\bm{u}^-$, and $\\bm{n}$, denoting the solution on \nthe exterior and interior of the element, and the outward-pointing normal vector, \nrespectively. Then, the discontinuous \nGalerkin method reads:\\\\\n\\indent\nFind $\\bm{u}_h \\in \\bm{V}_h^m$ such that, for all $\\bm{v}_h \\in \\bm{V}_h^m$,\n\\begin{gather} \n    \\label{eq:semi-disc-dg}\n    \\int_{K_i} \\partial_t \\bm{u}_h \\cdot \\bm{v}_h ~dx -\n        \\int_{K_i} \\bm{F}(\\bm{u}_h) : \\nabla \\bm{v}_h~dx\n        + \\oint_{\\partial K_i} \\widehat{\\bm{F}}(\\bm{u}^+, \\bm{u}^-, \\bm{n}) \\cdot \\bm{v}_h~ds = 0.\n\\end{gather}\n\n\\subsection{Advection equation}\nAs a first example, we consider the two-dimensional scalar advection equation\n\\begin{equation}\n    \\label{eq:2d-advection}\n    u_t + \\nabla \\cdot \\left( \\bm{\\beta} u \\right) = 0,\n\\end{equation}\nfor a given (constant) velocity vector $\\bm{\\beta} = (\\alpha, \\beta)$. We solve \nthis equation in the domain $[0, 2\\pi] \\times [0, 2\\pi]$, with periodic boundary \nconditions. The exact solution to this equation is given by\n\\begin{equation}\n    u(t, x, y) = u_0(x - \\alpha t, y - \\beta t),\n\\end{equation}\nwhere $u_0$ is the given initial state.\n\nIn order to define the discontinuous Galerkin method for equation \\eqref{eq:2d-advection}, \nwe define the \\emph{upwind flux} by\n\\begin{equation}\n    \\widehat{\\bm{F}}(\\bm{u}^+, \\bm{u}^-, \\bm{n}) =\n    \\begin{cases}\n        \\bm{u}^- \\quad\\text{if $\\bm{\\beta}\\cdot\\bm{n} \\geq 0$}\\\\\n        \\bm{u}^+ \\quad\\text{if $\\bm{\\beta}\\cdot\\bm{n} < 0$}\\\\\n    \\end{cases}\n\\end{equation}\nWe represent the approximate solution function $\\bm{u}_h$ as a vector $\\bm{U}$\nconsisting of the coefficients of the expansion of $\\bm{u}_h$ in terms of an\northogonal Legendre polynomial modal basis of the function space\n$\\bm{V}_h^m$. Discretizing equation \\eqref{eq:2d-advection} results in a linear\nsystem of equations, which we can write as\n\\begin{equation}\n    \\label{eq:adv-lin-system}\n    \\mathbf{M}(\\partial_t\\bm{U}) + \\mathbf{L}\\bm{U} = 0,\n\\end{equation}\nwhere the mass matrix $\\mathbf{M}$ corresponds to the first term on the left-hand \nside of \\eqref{eq:semi-disc-dg}, and $\\mathbf{L}$ consists of the second two \nterms on the left-hand side. The mass matrix is block-diagonal, and the matrix \n$\\mathbf{L}$ is a block matrix, with blocks along the diagonal, and off-diagonal \nblocks corresponding to the boundary terms from the neighboring elements.\n\n\\subsection{Temporal integration and linear solvers}\nWe consider the solution of \\eqref{eq:adv-lin-system} by means of implicit time \nintegration schemes, the simplest of which is the standard backward Euler scheme,\n\\begin{equation} \n    \\label{eq:be}\n    (\\mathbf{M} + k \\mathbf{L})\\bm{U}^{n+1} =  \\mathbf{M}\\bm{U}^n.\n\\end{equation}\nFurthermore, each stage of a higher-order scheme, such as a diagonally-implicit \nRunge-Kutta (DIRK) scheme \\cite{Alexander1977}, can be written as a \nsimilar equation. The block sparse system can be solved efficiently by means of\nan iterative linear solver. In this paper, we consider two solvers: the \nsimple block Jacobi method, and the preconditioned GMRES method.\n\n\\subsubsection{Block Jacobi method}\nA popular and simple iterative solver is the block Jacobi method, defined as\nfollows. Each iteration of the method for solving the linear system\n$\\mathbf{A}\\bm{x} = \\bm{b}$ is given by\n\\begin{equation}\n    \\label{eq:jacobi}\n    \\bm{x}^{(n+1)} = \\mathbf{D}^{-1}\\bm{b} + \\mathbf{R}_J\\bm{x}^{(n)},\n\\end{equation}\nwhere $\\mathbf{D}$ is the block-diagonal part of $\\mathbf{A}$, and $\\mathbf{R}_J\n= \\mathbf{I} - \\mathbf{D}^{-1}\\mathbf{A}$. This simple method has the advantage\nthat it is possible to analyze the convergence properties of the method simply\nby examining the eigenvalues of the matrix $\\mathbf{R}_J$. An upper bound of 1 \nfor the absolute value of the eigenvalues of the matrix $\\mathbf{R}_J$ is a \nnecessary and sufficient condition in order for Jacobi's method to converge \n(for any choice of initial vector $\\bm{x}^{(0)}$). The spectral radius of\n$\\mathbf{R}_J$ determines the speed of convergence.\n\n\\subsubsection{Preconditioned GMRES method}\nAnother popular and oftentimes more efficient \\cite{Bassi2000} method for solving\nlarge, sparse linear systems is the GMRES (generalized minimal residual) method\n\\cite{Persson2009}. As with most Krylov subspace methods, the choice of\npreconditioner has great impact on the efficiency of the solver\n\\cite{Persson-ILU0}. A simple and popular choice of preconditioner is the block\nJacobi preconditioner. Each application of this preconditioner is performed by\nmultiplying by the inverse of the block-diagonal part of the matrix. Another,\noften more effective choice of preconditioner is the block ILU(0) preconditioner\n\\cite{Diosady2009}. This preconditioner produces an approximate block-wise LU\nfactorization, whose sparsity pattern is enforced to be the same as that of the\noriginal matrix. This factorization can be performed in-place, and requires no\nmore storage that the original matrix. Unlike the block Jacobi method, the block\nILU(0) preconditioner can be highly sensitive to the ordering of the mesh\nelements \\cite{Duff1989,Benzi1999}. Because of this property, it is common to combine the use\nof ILU preconditioners with certain orderings of the mesh elements designed to\nincrease efficiency, such as reverse Cuthill-McKee \\cite{Cuthill1969}, minimum\ndegree \\cite{Markowitz1957}, nested dissection \\cite{George1973}, or minimum\ndiscarded fill \\cite{Persson2009}.\n\nIn this paper, we focus our study on the block Jacobi method, which is simpler\nand more amenable to analysis. We then perform numerical experiments using both\nthe block Jacobi method and the preconditioned GMRES method using ILU(0) and\nblock Jacobi preconditioning.\n\n\\section{Jacobi Analysis}\n\nWe compare tessellations of the plane by four sets of \\textit{generating\npatterns}, each consisting of one or more polygons. We consider tessellations\nconsisting of squares, regular hexagons, two right triangles, and two\nequilateral triangles. The generating patterns considered are shown in Figure\n\\ref{fig:gen-pat}. Each generating pattern $G_j$ consists of one or two\nelements, labeled $K_j$ and $\\widetilde{K_j}$. We will refer to these generating\npatterns as $S, H, R,$ and $E$ for squares, hexagons, right triangles, and\nequilateral triangles, respectively.\n\n\\begin{figure}[h]\n    \\centering\n    \\hspace*{\\fill}%\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{quad.tikz}\n        \\caption{Square Cartesian grid}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{hex.tikz}\n        \\caption{Regular hexagons}\n    \\end{subfigure}%\n    \\hspace*{\\fill}%\n    \n    \\hspace*{\\fill}%\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{rtri.tikz}\n        \\caption{Isosceles right triangles}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{etri.tikz}\n        \\caption{Equilateral triangles}\n    \\end{subfigure}\n    \\hspace*{\\fill}%\n    \n    \\caption{Examples of generating patterns $G_j$ shown with bolded lines. \n    Neighboring elements are shown unbolded.}\n    \\label{fig:gen-pat}\n\\end{figure}\n\nWe are interested in computing the spectral radius of the Jacobi matrix\n$\\mathbf{R_J}$ that arises from the discontinuous Galerkin discretization on the\nmesh resulting from tessellating the plane by each of the four generating\npatterns. For the sake of comparison, we choose the elements from each of the\ngenerating patters to have the same area. Therefore, if the side length of the\nequilateral triangle is $h_E = h$, then the two equal sides of the isosceles\nright triangle have side length $h_R = \\frac{\\sqrt[4]3}{\\sqrt 2}h_E$, the\nhexagon has side length $h_H = \\frac{1}{\\sqrt 6}h_E$, and the square has side\nlength $h_S = \\frac{\\sqrt[4]{3}}{2}h_E$. Then, the global system will have the\nsame number of degrees of freedom regardless of choice of generating pattern.\n\n\\subsection{Von Neumann analysis}\nFirst, we compare the efficiency of each of the four types of generating patterns when\nused to solve the advection equation \\eqref{eq:2d-advection} with the\ndiscontinuous Galerkin spatial discretization and implicit time integration. We\ncompute the spectral radius of the matrix $\\mathbf{R_J}$ using the classical von\nNeumann analysis for each of the generating patterns, in a manner similar to \n\\cite{Kubatko2008}.\n\nLet $\\bm{U}$ denote the solution vector, and let its $j$\\nth component,\n$\\bm{U}_j$, which is itself a vector, denote the degrees of freedom in $G_j$,\nthe $j$\\nth generating pattern. We remark that in the case of squares and\nhexagons, this corresponds exactly to the degrees of freedom in the element\n$K_j$, but in the case of the triangular generating patters, this corresponds to\nthe degrees of freedom from both of the elements $K_j$ and $\\widetilde{K_j}$. In\norder to determine the eigenvalues of $\\mathbf{R_J}$, we consider the planar\nwave with wavenumber $(n_x, n_y)$ defined by\n\\begin{equation}\n    \\bm{U}_j = e^{i(n_x x_j + n_y y_j)} \\widehat{\\bm{U}},\n\\end{equation}\nwhere $(x_j, y_j)$ are fixed coordinates in\n$G_j$. Then, we let $\\ell$ index the\ngenerating patterns neighboring $G_j$, and we let $\\bm{\\delta}_\\ell =\n(\\delta_{x\\ell}, \\delta_{y\\ell}) = (x_j - x_\\ell, y_j - y_\\ell)$\nbe the offsets satisfying $G_j + \\bm{\\delta}_\\ell = G_\\ell$. We can then\nwrite the solution in each of the neighboring generating patterns as\n\\begin{equation}\n    \\bm{U}_\\ell = e^{i(n_x \\delta_{x\\ell} + n_y \\delta_{y\\ell})} \\bm{U}_j.\n\\end{equation}\nIn this case we write the semi-discrete equations \\eqref{eq:adv-lin-system} in\nthe following compact form\n\\begin{equation}\n    \\mathbf{M}_j(\\partial_t \\bm{U}_j) + \n    \\sum_\\ell e^{i(n_x \\delta_{x\\ell} + n_y \\delta_{y\\ell})} \\mathbf{L}_{j\\ell} \\bm{U}_j = 0,\n\\end{equation}\nwhere the summation over $\\ell$ ranges over all neighboring generating patterns,\n$\\mathbf{M}_j$ denotes the diagonal block of $\\mathbf{M}$ \ncorresponding to the $j$th generating pattern, and\n$\\mathbf{L}_{j\\ell}$ denotes the block of $\\mathbf{L}$ in the $j$\\nth row and\n$\\ell$\\nth column. We can write\n\\begin{equation}\n    \\widehat{\\mathbf{L}}_j = \\sum_\\ell e^{i(n_x \\delta_{x\\ell} + n_y \\delta_{y\\ell})} \\mathbf{L}_{j\\ell}\n\\end{equation}\nto further simplify and obtain\n\\begin{equation}\n    \\label{eq:compact-form}\n    \\mathbf{M}_j(\\partial_t \\widehat{\\bm{U}}) + \\widehat{\\mathbf{L}}_j \\widehat{\\bm{U}} = 0.\n\\end{equation}\nIn order to solve equation \\eqref{eq:compact-form} using an implicit method, we\nconsider the backward Euler-type equation\n\\begin{equation}\n    (\\mathbf{M}_j + k \\widehat{\\mathbf{L}}_j) \\widehat{\\bm{U}}^{n+1} \n        = \\mathbf{M}_j \\widehat{\\bm{U}}^n.\n\\end{equation}\nThe Jacobi iteration matrix $\\mathbf{R_J}$ can then be written as\n\\begin{equation}\n    \\label{eq:compact-jacobi}\n    \\widehat{\\mathbf{R_J}}_j \n        = \\mathbf{I} - \\mathbf{D}^{-1}(\\mathbf{M}_j + k \\widehat{\\mathbf{L}}_j),\n\\end{equation}\nwhere the matrix $\\mathbf{D} = \\mathbf{M}_j + k \\mathbf{L}_{jj}$ consists of the\n$j$\\nth diagonal block of $\\mathbf{M} + k \\mathbf{L}$. The eigenvalues of the\nmatrix $\\widehat{\\mathbf{R_J}}_j$ control the speed of convergence of Jacobi's\nmethod. In the simple cases of piecewise constant functions ($p=0$), or in the\ncase of a one-dimensional domain, the eigenvalues can be computed explicitly. In\nthe more complicated case of $p \\geq 1$ in a two-dimensional domain, we compute\nthe eigenvalues numerically.\n\n\\subsection{1D example}\\label{sec:1d}\nTo illustrate the von Neumann analysis, we consider the one-dimensional scalar\nadvection equation\n\\begin{equation}\n    u_t + u_x = 0\n\\end{equation}\non the interval $[0, 2\\pi]$ with periodic boundary conditions. We divide the\ndomain into $N$ subintervals $K_j$, each of length $h$. Let $\\bm{U}$ denote the\nsolution vector, and let $\\bm{U}_j$ denote the degrees of freedom for the\n$j$\\nth interval $K_j$. For example, if piecewise constants are used, the method\nis identical to the upwind finite volume method, and each $\\bm{U}_j$ represents\nthe average of the solution over the interval. If piecewise polynomials of\ndegree $p$ are used, each $\\bm{U}_j$ is a vector of length $p+1$.\n\nFor the purposes of illustration, we choose $p=1$, and let $\\bm{U}_j = (u_{j,1},\nu_{j,2})$ represent the value of the solution at the left and right endpoints of\nthe interval $K_j$. Then, the local basis on the interval $K_j$ consists of the\nfunctions \n\\begin{equation}\n    \\label{eq:1d-basis}\n    \\phi_{j,1}(x) = j - x\/h, \\hspace{1in} \\phi_{j,2}(x) = x\/h - j + 1.\n\\end{equation}\nWe remark that the upwind flux in this case is always equal to the value of the\nfunction immediately to the left of the boundary point:\n\\begin{equation}\n    \\left[ \\widehat{\\bm{F}}(u^+, u^-, x) v(x) \\right]_{(j-1)h}^{jh}\n        = u_{j,2}v_{j,2} - u_{j-1,2}v_{j,1}.\n\\end{equation}\n\nThe entries of the $j$\\nth block of the mass matrix $\\mathbf{M}$ are given by\n\\begin{equation}\n    (\\mathbf{M}_j)_{i\\ell} = \\int_{(j-1)h}^{jh} \\phi_{j,i}(x) \\phi_{j,\\ell}(x)~dx.\n\\end{equation}\nAdditionally, we remark that the diagonal blocks of $\\mathbf{L}$ consist of the \nvolume integrals and right boundary terms given by\n\\begin{equation}\n    (\\mathbf{L}_{jj})_{i\\ell} = \\phi_{j,i}(jh)\\phi_{i,\\ell}(jh) \n        - \\int_{(j-1)h}^{jh} \\phi'_{j,i}(x)\\phi_{j,\\ell}(x)~dx.\n\\end{equation}\nWe let $\\bm{A}$ denote the backward Euler-type operator defined by\n\\begin{equation}\n    \\mathbf{A} = \\mathbf{M} + k\\mathbf{L},\n\\end{equation}\nand, solving the equation $\\mathbf{A}\\bm{x} = \\bm{b}$ by means of Jacobi iterations, \nwe define the Jacobi matrix $\\mathbf{R_J}$ by\n\\begin{equation}\n    \\mathbf{R_J} = \\mathbf{I} - \\mathbf{D}^{-1}\\mathbf{A},\n\\end{equation}\nwhere $\\mathbf{D}$ is the matrix consisting of the diagonal blocks of $\\mathbf{A}$. \nThe entries of the diagonal blocks $\\mathbf{M}_j$ and $\\mathbf{L}_{jj}$ can be computed \nexplicitly using \\eqref{eq:1d-basis} to obtain\n\\begin{equation}\n    \\mathbf{M}_j = \\left(\\begin{array}{cc}\n    \\frac{h}{3} & \\frac{h}{6} \\\\\n    \\frac{h}{6} & \\frac{h}{3}\n    \\end{array}\\right), \\quad\n    \\mathbf{L}_{jj} = \\left(\n    \\begin{array}{cc}\n     \\frac{1}{2} & \\frac{1}{2} \\\\\n     -\\frac{1}{2} & \\frac{1}{2} \\\\\n    \\end{array}\n    \\right), \\quad\n    \\mathbf{D}_j = \\left(\\begin{array}{cc}\n    \\frac{h}{3} + \\frac{k}{2} & \\frac{h}{6} + \\frac{k}{2} \\\\\n    \\frac{h}{6} - \\frac{k}{2} & \\frac{h}{3} + \\frac{k}{2}\n    \\end{array}\\right).\n\\end{equation}\n\nIn order to perform the von Neumann analysis, we seek solutions of the form \n$\\bm{U}_j = e^{inhj} \\widehat{\\bm{U}}$, which allows us to explicitly compute the \nform of the matrix $\\widehat{\\mathbf{L}}_j$. Recalling the compact form from \\eqref{eq:compact-form}, \nwe obtain\n\\begin{equation}\n    \\widehat{\\mathbf{L}}_j = \\left(\\begin{array}{cc}\n    \\frac{1}{2} & \\frac{1}{2}-e^{-ihn} \\\\\n    -\\frac{1}{2} & \\frac{1}{2}\n    \\end{array}\\right).\n\\end{equation}\nThen, the Jacobi matrix $\\widehat{\\mathbf{R_J}}_j$ is given by\n\\begin{equation}\n    \\widehat{\\mathbf{R_J}}_j =\n    \\left(\n\\begin{array}{cc}\n 0 & \\frac{2 e^{-i h n} k (2 h+3 k)}{h^2+4 k h+6 k^2} \\\\\n 0 & -\\frac{2 e^{-i h n} (h-3 k) k}{h^2+4 k h+6 k^2} \\\\\n\\end{array}\n\\right),\n\\end{equation}\nwhose eigenvalues $\\lambda_1$ and $\\lambda_2$ are given by\n\\begin{equation}\n    \\lambda_1 = 0, \\qquad \\lambda_2 = \\frac{2 k (3 k-h) e^{-i h n}}{h^2+4 h k+6 k^2}.\n\\end{equation}\nTherefore, each wavenumber $n$ from $0$ to $2\\pi\/h$ corresponds to an eigenvalue\nof the Jacobi matrix $\\mathbf{R_J}$, and the magnitude of these eigenvalues\ndetermine the speed of convergence of Jacobi's method. In this case, the\nexpression\n\\begin{equation}\n    \\lambda_{\\rm max} = \\frac{2 k \\left| h-3 k\\right| }{h^2+4 h k+6 k^2}\n\\end{equation}\ndetermines the speed of convergence of Jacobi's method. This expression can \neasily be seen to be bounded above by 1 for all positive values of $h$ and $k$, \ntherefore indicating that Jacobi's method is guaranteed to converge, \nunconditionally, regardless of spatial resolution or timestep. \n\n\\subsection{2D analysis}\\label{sec:2d-analysis}\nWe now turn to the analysis of the four generating patterns shown in Figure \n\\ref{fig:gen-pat}. The analysis proceeds along the same lines as in the \none-dimensional example from Section \\ref{sec:1d}. As an example, we present \nthe case of piecewise constants, for which it is possible to explicitly compute \nthe eigenvalues of the Jacobi matrix $\\mathbf{R_J}$. In this case the \ndiscontinuous Galerkin formulation simplifies to the upwind finite volume method\n\\begin{equation}\n    \\int_{K_j} \\partial_t u_h~dx  \n        + \\oint_{\\partial K_j} \\widehat{\\bm{F}}(u^+, u^-, \\bm{n})~ds = 0.\n\\end{equation}\nFor the sake of concreteness, we assume without loss of generality\nthat the velocity vector \n$\\bm{\\beta} = (\\alpha, \\beta)$ satisfies $\\alpha, \\beta \\geq 0$. In order to \nexplicitly write the upwind flux on the meshes \nconsisting of hexagons and equilateral triangles, we further assume that \n$\\sqrt{3}\\alpha - \\beta \\geq 0$, and on the mesh consisting of right triangles \nwe assume that $\\alpha - \\beta \\geq 0$.\nIn the case of the square and hexagonal meshes, there \nis only one degree of freedom per generating pattern, and we will write \n$u_j$ to represent the average value of the solution over the generating \npattern $G_j$. We then consider the planar wave with wavenumber \n$(n_x, n_y)$ given by $u_j = e^{i(n_x x_j + n_y y_j)} \\widehat{u}$. \nIn the case of the square mesh with side length $h_S =\n\\frac{\\sqrt[4]{3}}{2}h_E$, the method can be written as\n\\begin{equation}\n    h_S^2 \\left( \\partial_t \\widehat{u} \\right) = \n        - h_S \\left( \n            \\alpha(1  - e^{-i n_x h_S} )\n            + \\beta( 1 - e^{-i n_y h_S} ) \\right) \\widehat{u}.\n\\end{equation}\nIn this case, the mass matrix $\\mathbf{M}$ is a diagonal matrix with $h_S^2$\nalong the diagonal, and the diagonal entries of the matrix $\\mathbf{L}$ are\ngiven by $h_S(\\alpha + \\beta)$. Therefore, the eigenvalues of the Jacobi matrix\n$\\mathbf{R_J^S} = \\mathbf{I} - D^{-1}(\\mathbf{M} + k\\mathbf{L})$ are given by\n\\begin{equation}\n\\begin{aligned}\n    \\label{eq:fv-sq-eigs}\n    \\lambda(\\mathbf{R_J^S}) &= 1 - \\frac{1}{h_S^2 + h_Sk(\\alpha + \\beta)}\n        \\left(h_S^2 + h_Sk\\left( \\alpha(1  - e^{-i n_x h_S} )\n            + \\beta( 1 - e^{-i n_y h_S} ) \\right) \\right) \\\\\n        &= \\frac{k\\left( \\alpha e^{-in_x h_S} + \\beta e^{-in_y h_S} \\right)}\n                 {h_S + k(\\alpha+\\beta)}.\n\\end{aligned}\n\\end{equation}\n\nIn the case of the hexagonal mesh with side length $h_H =\n\\frac{1}{\\sqrt{6}}h_E$, the method is\n\\begin{equation}\n\\begin{aligned}\n    \\frac{3\\sqrt{3}}{2}h_H^2 \\left(\\partial_t \\widehat{u} \\right) &= \n    - h_H\\Bigg(\n    \\left( \\sqrt{3}\\alpha + \\beta \\right)\n    +\\left( - \\tfrac{\\sqrt{3}}{2}\\alpha + \\tfrac{\\beta}{2} \\right)e^{ih_H\\left(-\\frac{3}{2}n_x + \\frac{\\sqrt{3}}{2}n_y\\right)}\\\\\n    &\\qquad\n    +\\left( - \\tfrac{\\sqrt{3}}{2}\\alpha - \\tfrac{\\beta}{2} \\right)e^{ih_H\\left(-\\frac{3}{2}n_x - \\frac{\\sqrt{3}}{2}n_y\\right)}\n    -\\beta e^{-ih_H\\sqrt{3}n_y}\n    \\Bigg) \\widehat{u}.\n\\end{aligned}\n\\end{equation}\nA similar analysis shows that the eigenvalues of the matrix $\\mathbf{R_J^H}$ are \ngiven by\n\\begin{equation}\n\\begin{aligned}\n    \\label{eq:fv-hex-eigs}\n    \\lambda(\\mathbf{R_J^H})\n    &=\\tfrac{k e^{-\\frac{1}{2} i h_H \\left(3 n_x+\\sqrt{3} n_y\\right)} \n                \\left(\\sqrt{3} \\beta \\left(2\n                    e^{\\frac{1}{2} i h_H \\left(3 n_x-\\sqrt{3} n_y\\right)}\n                    - e^{i \\sqrt{3} h_H n_y}+1\\right)+\n                3 \\alpha  \\left(1+e^{i \\sqrt{3} h_H n_y}\\right)\\right)}\n            {9 h_H + 6 \\alpha  k+2 \\sqrt{3} \\beta  k}.\n\\end{aligned}\n\\end{equation}\n\nIn the case of the two triangular meshes, there are two degrees of freedom per \ngenerating pattern, corresponding to the elements $K_j$ and $\\widetilde{K_j}$ \nin the generating pattern $G_j$. We write $\\bm{U}_j = (u_{j,1}, u_{j,2})$, \nwhere $u_{j,1}$ is the average of the solution over the element $K_j$, and \n$u_{j,2}$ is the average of the solution over $\\widetilde{K_j}$. The planar \nwave solution is then given by $\\bm{U}_j = e^{i(n_x x_j + n_y y_j)} \\widehat{\\bm{U}}$, \nfor $\\widehat{\\bm{U}} = (\\widehat{u}_1, \\widehat{u}_2)$. We consider the case of a \nright-triangular mesh, where the two equal sides of the isosceles right triangles \nhave length $h_R = \\frac{\\sqrt[4]{3}}{\\sqrt{2}}h_E$. The method then reads:\n\\begin{equation}\n\\partial_t \\left(\\begin{array}{c} \\widehat{u}_1 \\\\ \\widehat{u}_2 \\end{array}\\right)\n= -\\frac{2}{h_R}\\left(\\begin{array}{c}\n     \\alpha \\widehat{u}_1 - e^{-ih_Rn_x} \\alpha \\widehat{u}_2  \\\\\n     \\alpha \\widehat{u}_2 + (\\beta - \\alpha) \\widehat{u}_1 \n        - e^{-ih_Rn_y} \\beta \\widehat{u}_1 \n\\end{array}\\right).\n\\end{equation}\nIn the case of the mesh consisting of equilateral triangles, each with side\nlength $h_E$, the method reads:\n\\begin{equation}\n\\partial_t \\left(\\begin{array}{c} \\widehat{u}_1 \\\\ \\widehat{u}_2 \\end{array}\\right)\n= \\tfrac{-4}{\\sqrt{3}h_E}\\left(\\begin{array}{c}\n    \\left(\\frac{\\sqrt 3}{2}\\alpha \n        + \\frac{1}{2}\\beta\\right)\\widehat{u}_1\n    + \\left(e^{-ih_En_x}\\left(-\\frac{\\sqrt 3}{2}\\alpha + \\frac{1}{2}\\beta \\right)\n    - e^{-ih_En_y} \\beta\\right) \\widehat{u}_2 \\\\\n    %\n    %\n    \\left(-\\frac{\\sqrt 3}{2}\\alpha \n    - \\frac{1}{2}\\beta\\right) \\widehat{u}_1 + \\left(\\frac{\\sqrt 3}{2}\\alpha \n    + \\frac{1}{2}\\beta \\right) \\widehat{u}_2\n\\end{array}\\right).\n\\end{equation}\nComputing the eigenvalues of the corresponding Jacobi matrices $\\mathbf{R_J^R}$ \nand $\\mathbf{R_J^E}$, we obtain\n\\begin{align}\n    \\label{eq:fv-rt-eigs}\n    \\lambda(\\mathbf{R_J^R}) &= \\pm\\frac{2 k e^{-\\frac{1}{2} i h_R (n_x+n_y)} \n        \\sqrt{\\alpha } \\sqrt{\\beta +(\\alpha -\\beta ) e^{i h_R n_y}}}{h_R+2 \\alpha  k}, \\\\\n    \\label{eq:fv-et-eigs}\n    \\lambda(\\mathbf{R_J^E}) &= \\pm \\frac{2 k \\left(3 \\alpha +\\sqrt{3} \\beta \\right) \\sqrt{2 \\beta  e^{i h_E n_x}+\\left(\\sqrt{3} \\alpha -\\beta \\right) e^{i h_E n_y}}}{\\left(3 h_E+6 \\alpha  k+2 \\sqrt{3} \\beta  k\\right) \\sqrt{\\left(\\sqrt{3} \\alpha +\\beta \\right) e^{i h_E (n_x+n_y)}}}.\n\\end{align}\n\nThen, equations \\eqref{eq:fv-sq-eigs}, \\eqref{eq:fv-hex-eigs},\n\\eqref{eq:fv-rt-eigs}, and \\eqref{eq:fv-et-eigs} completely determine the speed\nof convergence for Jacobi's method of each of the four generating patterns\nconsidered. In the case of a higher-order discontinuous Galerkin method with\nbasis consisting of piecewise polynomials of degree $p>0$, we obtain a Jacobi\nmatrix given by equation \\eqref{eq:compact-jacobi}, where the matrices\n$\\widehat{\\mathbf{R_J}}_j, \\mathbf{D}, \\mathbf{M}_j,$ and\n$\\widehat{\\mathbf{L}}_j$ are $\\frac{(p+1)(p+2)}{2}\\times\\frac{(p+1)(p+2)}{2}$\nblocks. In this case, we do not obtain closed-form expressions for the\neigenvalues, but rather compute them numerically.\n\nWe normalize the velocity magnitude and consider $\\bm{\\beta} = (\\cos(\\theta),\n\\sin(\\theta))$. On the square mesh, $\\theta$ can range from $0$ to $\\pi\/2$. On\nthe hexagonal and equilateral triangle meshes, $\\theta$ ranges from $0$ to\n$\\pi\/3$, and on the right-triangular mesh $\\theta$ ranges from $0$ to $\\pi\/4$.\nWe consider a fixed spatial resolution $h$, and compare the efficiency of the\nfour patterns for three choices of temporal resolution. We first consider an\n``explicit'' time step, satisfying the CFL-type condition\n\\begin{equation}\n    \\label{eq:cfl}\n    k_{\\rm exp} = \\frac{h}{|\\bm{\\beta}|}.\n\\end{equation}\nAs one advantage of using an implicit method is that we are not limited by an\nexplicit timestep restriction of the form \\eqref{eq:cfl}, we consider three implicit time\nsteps given by $k_1 = 3 k_{\\rm exp}$, $k_2 = 2 k_1$, and $k_3 = 4 k_1$. We\nthen maximize over a discrete sample of $\\theta \\in [0, \\pi\/4]$ and over all\nwavenumbers $(n_x, n_y)$, in order to compute maximum eigenvalue for each of the\ngenerating patterns. As the number of iterations required to converge to a given\ntolerance scales like the reciprocal of the logarithm of the spectral radius, we\ncompare the efficiency of the generating patterns by considering the ratio\n\\[ \\frac{\\log\\left(\\lambda_{\\rm max}(\\mathbf{R_J^{\\mathrm{min}}})\\right)}\n        {\\log\\left(\\lambda_{\\rm max}(\\mathbf{R_J^*})\\right)}, \\]\nwhere $\\lambda_{\\rm max}(\\mathbf{R_J^*})$ is the largest eigenvalue of \n$\\mathbf{R_J^*}$, for $* = H, S, R, E$, and \n$\\lambda_{\\rm max}(\\mathbf{R_J^{\\mathrm{min}}})$ is the smallest among all \n$\\lambda_{\\rm max}(\\mathbf{R_J^*})$. This ratio corresponds to the ratio of\niterations required to converge to a given tolerance when compared with the most\nefficient among the generating patterns. The results obtained for $p=0,1,2,3$,\nand $k=k_1,k_2,k_3$ for each generating pattern are shown in Table \\ref{tab:eig}\nand Figure \\ref{fig:bars}.\n\n\\newcommand{\\cellcolor[gray]{0.9}\\bf}{\\cellcolor[gray]{0.9}\\bf}\n\\begin{table}[!t]\n    \\centering\n    \\tiny\n       \\begin{tabular}{r | lll | lll}\n            & \\multicolumn{3}{c|}{\\small $p=0$} & \\multicolumn{3}{c}{\\small $p=1$} \\\\\n            & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ \\\\\n            \\hline\n            Hexagons    &\n\\highlightcell1.000000 & \\highlightcell1.000000 & \\highlightcell1.000000 & \n\\highlightcell1.000000 & \\highlightcell1.000000 & \\highlightcell1.000000\\\\\n            Squares                 & \n1.128939    &   1.133989    &   1.136772    &\n1.058098\t&\t1.118222\t&\t1.130101\\\\\n            Right triangles         & \n1.128939\t&\t1.133989\t&\t1.136772    &\n1.084223\t&\t1.132326\t&\t1.137313\\\\\n            Equilateral triangles   & \n1.207328\t&\t1.215467\t&\t1.219948    &\n1.137267\t&\t1.201638\t&\t1.214376\n        \\end{tabular}\n       \\vspace{8pt}\n   \\vspace{12pt}\n       \\begin{tabular}{r | lll | lll}\n            & \\multicolumn{3}{c|}{\\small $p=2$} & \\multicolumn{3}{c}{\\small $p=3$} \\\\\n            & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ \\\\\n            \\hline\n            Hexagons                &\n\\highlightcell1.000000 & \\highlightcell1.000000 & \\highlightcell1.000000 &\n1.077183\t&\t1.070785\t&\t1.066101\\\\\n            Squares                 & \n1.095785\t&\t1.118510\t&\t1.129314    &\n\\highlightcell1.000000  &   \\highlightcell1.000000  &   \\highlightcell1.000000\\\\\n            Right triangles         & \n1.111863\t&\t1.126951\t&\t1.133634    &\n1.010482\t&\t1.005391\t&\t1.002733\\\\\n            Equilateral triangles   & \n1.177503\t&\t1.201918\t&\t1.213527    &\n1.074570\t&\t1.074570\t&\t1.074570\n        \\end{tabular}\n    \\vspace{8pt}\n   \\caption{Ratio of logarithm of eigenvalues\n   $\\log\\left(\\lambda_{\\rm max}(\\mathbf{R_J^{\\mathrm{min}}})\\right)\n   \/\\log\\left(\\lambda_{\\rm max}(\\mathbf{R_J^*})\\right)$ ranging \n   over angle $\\theta$ and wavenumber $(n_x, n_y)$, for piecewise polynomials \n   of degree 0, 1, 2, and 3, for varying choices of time step $k$. The smallest\n   eigenvalue in each column is highlighted.}\n    \\label{tab:eig}\n\\end{table}\n\\begin{figure}[!b]\n   \\centering\n   \\hspace*{\\fill}%\n   \\begin{subfigure}{0.3\\textwidth}\n       \\centering\n           \\begin{tikzpicture}\n           \\begin{axis}[\n             ymin=0.75,\n             ymax=1.35,\n             width=1.95in,\n             ybar=0pt\n             bar width=7pt,\n             xtick=data,\n             every axis\/.append style={font=\\tiny},\n             xticklabels={$k_1$, $k_2$, $k_3$},\n             enlarge x limits=0.3,\n             major x tick style = {opacity=0},\n             legend cell align=left,\n             legend entries={Hexagons,\n                             Squares,\n                             Right Triangles,\n                             Equilateral Triangles},\n             legend columns=4,\n             legend style={\n                draw=none,\n                \/tikz\/every even column\/.append style={column sep=0.5cm}},\n             legend to name=leg:barlegend\n          ]\n          \\addplot table[header = false, x index = 0, y index = 1] {deg0.dat};\n          \\addplot table[header = false, x index = 0, y index = 2] {deg0.dat};\n          \\addplot table[header = false, x index = 0, y index = 3] {deg0.dat};\n          \\addplot table[header = false, x index = 0, y index = 4] {deg0.dat};\n          \\end{axis}\n          \\end{tikzpicture}\n      \\caption{$p=0$}\n  \\end{subfigure\n  \\begin{subfigure}{0.3\\textwidth}\n      \\centering\n          \\begin{tikzpicture}\n          \\begin{axis}[\n             ymin=0.75,\n             ymax=1.35,\n             width=1.95in,\n             ybar=0pt\n             bar width=7pt,\n             xtick=data,\n             every axis\/.append style={font=\\tiny},\n             xticklabels={$k_1$, $k_2$, $k_3$},\n             enlarge x limits=0.3,\n             major x tick style = {opacity=0},\n          ]\n          \\addplot table[header = false, x index = 0, y index = 1] {deg1.dat};\n          \\addplot table[header = false, x index = 0, y index = 2] {deg1.dat};\n          \\addplot table[header = false, x index = 0, y index = 3] {deg1.dat};\n          \\addplot table[header = false, x index = 0, y index = 4] {deg1.dat};\n          \\end{axis}\n          \\end{tikzpicture}\n      \\caption{$p=1$}\n  \\end{subfigure}%\n \n  \\hspace*{\\fill}%\n  \n  \\hspace*{\\fill}%\n  \\begin{subfigure}{0.3\\textwidth}\n      \\centering\n          \\begin{tikzpicture}\n          \\begin{axis}[\n             ymin=0.75,\n             ymax=1.35,\n             width=1.95in,\n             ybar=0pt\n             bar width=7pt,\n             xtick=data,\n             every axis\/.append style={font=\\tiny},\n             xticklabels={$k_1$, $k_2$, $k_3$},\n             enlarge x limits=0.3,\n             major x tick style = {opacity=0},\n          ]\n          \\addplot table[header = false, x index = 0, y index = 1] {deg2.dat};\n          \\addplot table[header = false, x index = 0, y index = 2] {deg2.dat};\n          \\addplot table[header = false, x index = 0, y index = 3] {deg2.dat};\n          \\addplot table[header = false, x index = 0, y index = 4] {deg2.dat};\n          \\end{axis}\n          \\end{tikzpicture}\n      \\caption{$p=2$}\n  \\end{subfigure\n  \\begin{subfigure}{0.3\\textwidth}\n      \\centering\n          \\begin{tikzpicture}\n          \\begin{axis}[\n             ymin=0.75,\n             ymax=1.35,\n             width=1.95in,\n             ybar=0pt\n             bar width=7pt,\n             xtick=data,\n             every axis\/.append style={font=\\tiny},\n             xticklabels={$k_1$, $k_2$, $k_3$},\n             enlarge x limits=0.3,\n             major x tick style = {opacity=0},\n          ]\n          \\addplot table[header = false, x index = 0, y index = 1] {deg3.dat};\n          \\addplot table[header = false, x index = 0, y index = 2] {deg3.dat};\n          \\addplot table[header = false, x index = 0, y index = 3] {deg3.dat};\n          \\addplot table[header = false, x index = 0, y index = 4] {deg3.dat};\n          \\end{axis}\n          \\end{tikzpicture}\n      \\caption{$p=3$}\n  \\end{subfigure}\n \n  \\hspace*{\\fill}%\n  \n  \\tikzexternaldisable\\ref{leg:barlegend}\\tikzexternalenable\n  \\caption{Ratios of the logarithm of the largest eigenvalues for \n  each pattern.}\n  \\label{fig:bars}\n\\end{figure}\n\nWe remark that for degrees 0, 1, and 2 polynomials, the hexagonal mesh resulted\nin the smallest eigenvalues for all choices of timestep considered, and the\nsquare mesh resulted in the second-smallest eigenvalues. For degree 3\npolynomials, the square mesh resulted in the smallest eigenvalues for all cases\nconsidered. We notice a significant decrease in the expected performance of the\nhexagonal elements in the case of $p = 3$, although we have noticed that the \neffect observed in practice is not as significant as the theoretical results\nwould suggest.\n\n\\pagebreak\n\n\\section{Numerical Results}\\label{sec:numerical}\n\n\\subsection{Advection with variable velocity field} \\label{sec:variable-velocity}\nTo perform numerical experiments extending the analysis of equation\n\\eqref{eq:2d-advection} beyond the case of a constant velocity $\\bm{\\beta}$, we\nconsider a variable velocity field $\\bm{\\beta}(x, y)$. In this case, the upwind\nnumerical flux \n\\begin{equation}\n    \\widehat{\\bm{F}}(\\bm{u}^+, \\bm{u}^-, \\bm{n}, x, y) =\n    \\begin{cases}\n        \\bm{u}^-(x,y) \\quad\\text{if $\\bm{\\beta}(x, y)\\cdot\\bm{n} \\geq 0$}\\\\\n        \\bm{u}^+(x,y) \\quad\\text{if $\\bm{\\beta}(x, y)\\cdot\\bm{n} < 0$}\\\\\n    \\end{cases}\n\\end{equation}\nis evaluated point-wise. As an example, we define the velocity to be given by the\nvector field $\\bm{\\beta}(x,y) = (2y - 1, -2x + 1)$ on the spatial domain $\\Omega\n= [0,1]\\times[0,1]$. This velocity field is shown in Figure \\ref{fig:vel-field}.\nWe let the initial conditions be given by the Gaussian centered at $(x_0, y_0) =\n(0.35, 0.5)$,\n\\begin{equation}\n    u_0(x, y) = \\exp(-150((x-x_0)^2 + (y-y_0)^2)).\n\\end{equation}\nThe exact solution is periodic with period $\\pi$, and is given by the rotation\nabout the center of the domain,\n\\begin{equation}\n    u(x, y, t) = \\exp(-150((x - 0.5 + 0.15\\cos(2t))^2 + (y - 0.5 - 0.3\\cos(t)\\sin(t))^2)).\n\\end{equation}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{velfield}\n    \\caption{Velocity field $\\bm{\\beta}(x,y) = (2y - 1, -2x + 1)$}\n    \\label{fig:vel-field}\n\\end{figure}\n\n\\subsubsection{Convergence of the block Jacobi method}\nWe consider meshes of the domain created by repeating each of the four\ngenerating patterns considered in the previous section. As before, for fixed\nspatial resolution $h$, we choose $h_H, h_S, h_R,$ and $h_E$ such that the\nnumber of degrees of freedom is the same for each mesh. We then solve the\nadvection equation using the backward Euler time discretization, where the block\nJacobi iterative method is used to solve the resulting linear system.\nThe zero vector is used as the starting vector for the block\nJacobi solver.{ We choose $h = 0.05$, and since $\\max_{(x,y)} |\\bm{\\beta}(x,y)\n| = \\sqrt{2}$, we consider time steps of $k_1 = h\/\\sqrt{2}$, $k_2 = 2 k_1$, $k_3\n= 4 k_1$. The number of iterations required for the block Jacobi method to\nconverge to a tolerance of $10^{-14}$ are given in Table\n\\ref{tab:gaussian-iters}.\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n    \\hline\n    Hexagons                & \\highlightcell33 & \\highlightcell57 &\n        \\highlightcell104 & \\highlightcell21 & \\highlightcell41 &\n        \\highlightcell77  & 24 & \\highlightcell41 & \\highlightcell77 & \n        \\highlightcell21 & \\highlightcell39 & \\highlightcell75 \\\\\n    Squares                 & 35 & 61 & 109 & \\highlightcell21 & 42 & 83  \n        & \\highlightcell22 & 42 & 83  & 22 & 42 & 81 \\\\\n    Right triangles         & 39 & 68 & 128 & 26 & 51 & 100 & 25 & 51 & 100 & 25 & 51 & 100 \\\\\n    Equilateral triangles   & 37 & 67 & 123 & 25 & 47 & 92  & 25 & 47 & 92 & 24 & 47 & 91 \\\\\n    \\end{tabular}\n    \\vspace{12pt}\n    \\caption{Iterations required for the block Jacobi iterative method to \n        converge in the case of a non-constant velocity field. The smallest\n        number of iterations in each column is highlighted.}\n    \\label{tab:gaussian-iters}\n\\end{table}\n\nThe results are similar to those from the analysis performed in Section\n\\ref{sec:2d-analysis}. We note that the hexagonal and square meshes resulted in\nthe lowest number of Jacobi iterations for all of the test cases considered. In\ncontrast to the results of Section \\ref{sec:2d-analysis}, we do not observe a \ndecrease in the performance of the hexagonal elements for the case of $p=3$, \nand instead the performance is similar among all choices of $p$ considered.\n\n\\subsubsection{Randomly perturbed mesh}\nWe now consider the effect of polygonal elements on irregular meshes. To this\nend, we consider a set of \\emph{generating points} distributed evenly on a\nCartesian grid with mesh size $h$. Then, each point is perturbed by a random\nperturbation sampled uniformly from the interval $[-\\delta, \\delta]$. We obtain\ntwo randomized meshes by constructing the Delaunay triangulation and Voronoi\ndiagram resulting from this set of generating points. The Delaunay mesh consists\nentirely of triangular elements, whereas the Voronoi diagram is constructed out\nof arbitrary polygonal elements. Examples of the two meshes considered are shown\nin Figure \\ref{fig:random-mesh}. In contrast to the regular meshes considered in\nthe previous examples, these two meshes do not consist of the same number of\nelements. The Voronoi diagram consists of about half the number of elements as\nthe Delaunay triangulation. In the test case considered, the randomized\npolygonal mesh consists of 410 polygonal elements, whereas the randomized\ntriangular mesh consists of 759 triangular elements.\n\nThe governing equations and set-up is the same as in the previous section. We\nrecord the number of block Jacobi iterations required to converge to  a\ntolerance of $10^{-14}$\nin Table \\ref{tab:random-iters}. Because there is a difference in the number of\nmesh elements, the resulting linear system will have a different total number of\ndegrees of freedom. This difference will then have an additional effect on the\nspeed of convergence of the block Jacobi method. We note that for polynomials of\ndegree $p=0,1,2,3$ and for all choices of time step $k$ considered, solving the\nsystem resulting from the Voronoi diagram requires fewer block Jacobi iterations\nthan does solving the system resulting from the corresponding Delaunay\ntriangulation.\n\n\\begin{figure}[h]\n    \\centering\n    \\hspace*{\\fill}%\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\begin{tikzpicture}\n        \\begin{axis}[\n            width=\\textwidth,\n            height=\\textwidth,\n            no markers,\n        ]\n        \\addplot[black] table {randomtri.dat};\n        \\end{axis}\n        \\end{tikzpicture}\n        \\caption{Delaunay triangulation}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\begin{tikzpicture}\n        \\begin{axis}[\n            width=\\textwidth,\n            height=\\textwidth,\n            no markers,\n        ]\n        \\addplot[black] table {randompoly.dat};\n        \\end{axis}\n        \\end{tikzpicture}\n        \\caption{Voronoi diagram}\n    \\end{subfigure}%\n    \\hspace*{\\fill}%\n    \\caption{Randomized polygonal and triangular meshes corresponding to the same \n    set of generating points.}\n    \\label{fig:random-mesh}\n\\end{figure}\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n    \\hline\n    Voronoi diagram &\n        \\highlightcell27 & \\highlightcell32 & \\highlightcell38 &\n        \\highlightcell24 & \\highlightcell33 & \\highlightcell38  & \n        \\highlightcell24 & \\highlightcell32 & \\highlightcell36 & \n        \\highlightcell22 & \\highlightcell31 & \\highlightcell36 \\\\\n    Delaunay triangulation & \n        38 & 48 & 52 & \n        33 & 45 & 48  &\n        33 & 46 & 50 & \n        33 & 44 & 48 \\\\\n    \\end{tabular}\n    \\vspace{12pt}\n    \\caption{Iterations required for the block Jacobi iterative method to \n        converge in the case of irregular, randomly perturbed meshes. The\n        smallest number of iterations in each column is highlighted.}\n    \\label{tab:random-iters}\n\\end{table}\n\n\\subsubsection{Convergence of the GMRES method} \\label{sec:adv-gmres}\nThe above analysis focused on the block Jacobi method largely because of the\nsimplicity of the method. In practice, more sophisticated iterative methods are\noften used \\cite{Persson2009}. In this section, we consider the solution of the\nlinear system \\eqref{eq:be} by means of the GMRES method, using both the block\nJacobi and the block ILU(0) preconditioners. Since the computational work\nincreases per iteration in GMRES, we choose a \\textit{restart parameter} of 20\niterations \\cite{saad03iterative}.\nWe repeat the above test case of the advection equation with variable velocity \nfield and record the number of GMRES iterations required to converge to \na tolerance of $10^{-14}$ using the block Jacobi preconditioner \nin Table \\ref{tab:gmres-jac}.\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n    \\hline\n    Hexagons &\n        \\highlightcell31 & \\highlightcell53 & \\highlightcell92 &\n        \\highlightcell25 & \\highlightcell42 & \\highlightcell80 &\n        28 & \\highlightcell47 & \\highlightcell86 &\n        28 & \\highlightcell49 & \\highlightcell90 \\\\\n    Squares &\n        37 & 64 & 116 &\n        27 & 51 & 101 &\n        \\highlightcell27 & 51 & 98 & \n        \\highlightcell27 & 52 & 100 \\\\\n    Right triangles &\n        40 & 70 & 134 &\n        33 & 61 & 123 &\n        31 & 60 & 117 &\n        29 & 59 & 115\\\\\n    Equilateral triangles &\n        39 & 67 & 124 &\n        33 & 58 & 113 & \n        32 & 59 & 113 &\n        31 & 57 & 111 \\\\\n    \\end{tabular}\n    \\vspace{12pt}\n    \\caption{Iterations required for the GMRES iterative method with block \n    Jacobi preconditioner to converge. The smallest number of iterations in \n    each column is highlighted.}\n    \\label{tab:gmres-jac}\n\\end{table}\n\nWe now consider the solution of the above problem using the GMRES method with\nthe block ILU(0) preconditioner. Because of the sensitivity of the block ILU(0)\nfactorization to the ordering of the mesh elements, and for the sake of a fair\ncomparison between the generating patterns, we consider the \\textit{natural\nordering} of mesh elements, illustrated in Figure \\ref{fig:element-ordering}.\nAs in the case of the block Jacobi preconditioner, we repeat the test case of\nthe advection equation with variable velocity field. We record the number of\nGMRES iterations required to converge to the above tolerance using the block ILU(0)\npreconditioner in Table \\ref{tab:gmres-ilu}. In this case, the square mesh\nresulting in the smallest number of iterations in all of the trials. The mesh\nconsisting of right isosceles triangles resulted in the largest number of\niterations in all trials. We further note that the number of GMRES iterations \nrequired when using the block Jacobi preconditioner scales similarly to the \nnumber of block Jacobi iterations required, as recorded in Table \n\\ref{tab:gaussian-iters}. We note that the block ILU(0) preconditioner \nrequires fewer GMRES iterations to converge, and the number of iterations scales\nmore favorably in $k$, when compared with the block Jacobi \npreconditioner.\n\n\\begin{figure}[h]\n    \\centering\n    \\hspace*{\\fill}%\n    \\begin{subfigure}{0.4\\textwidth}\n        \\centering\n        \\begin{tikzpicture}\n            \\begin{axis}[\n                hide axis,\n                xmin=-0.35,\n                xmax=0.75,\n                axis equal,\n                width=\\textwidth,\n                height=\\textwidth,\n                no markers,\n                ticks=none,\n            ]\n            \\addplot[black] table {hex.dat};\n            \\node at (axis cs:-0.14, 0) {1};\n            \\node at (axis cs:0.14,  0) {2};\n            \\node at (axis cs:0.42,  0) {3};\n            \n            \\node at (axis cs:0,   0.25) {4};\n            \\node at (axis cs:0.28,0.25) {5};\n            \\node at (axis cs:0.56,0.25) {6};\n            \n            \\node at (axis cs:-0.14, 0.5) {7};\n            \\node at (axis cs:0.14,  0.5) {8};\n            \\node at (axis cs:0.42,  0.5) {9};\n            \\end{axis}\n        \\end{tikzpicture}\n        \\caption{Hexagonal mesh}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{0.4\\textwidth}\n        \\centering\n        \\begin{tikzpicture}\n            \\begin{axis}[\n                hide axis,\n                xmin=-0.15,\n                xmax=0.95,\n                axis equal,\n                width=\\textwidth,\n                height=\\textwidth,\n                no markers,\n                ticks=none,\n            ]\n            \\addplot[black] table {quad.dat};\n            \\node at (axis cs:0.13,  0.13) {1};\n            \\node at (axis cs:0.4,  0.13) {2};\n            \\node at (axis cs:0.65, 0.13) {3};\n            \n            \\node at (axis cs:0.13,  0.4) {4};\n            \\node at (axis cs:0.4,  0.4) {5};\n            \\node at (axis cs:0.65, 0.4) {6};\n            \n            \\node at (axis cs:0.13,  0.65) {7};\n            \\node at (axis cs:0.4,  0.65) {8};\n            \\node at (axis cs:0.65, 0.65) {9};\n            \\end{axis}\n        \\end{tikzpicture}\n        \\caption{Square mesh}\n    \\end{subfigure}%\n    \\hspace*{\\fill}%\n    \n    \\hspace*{\\fill}%\n    \\begin{subfigure}{0.4\\textwidth}\n        \\centering\n        \\begin{tikzpicture}\n            \\begin{axis}[\n                hide axis,\n                xmin=-0.15,\n                xmax=0.95,\n                axis equal,\n                width=\\textwidth,\n                height=\\textwidth,\n                no markers,\n                ticks=none,\n            ]\n            \\addplot[black] table {rtri.dat};\n            \\node at (axis cs:0.125,  0.248) {1};\n            \\node at (axis cs:0.248,  0.125) {2};\n            \\node at (axis cs:0.496,  0.248) {3};\n            \\node at (axis cs:0.640,  0.125) {4};\n            \n            \\node at (axis cs:0.125,  0.620) {5};\n            \\node at (axis cs:0.248,  0.496) {6};\n            \\node at (axis cs:0.496,  0.620) {7};\n            \\node at (axis cs:0.640,  0.496) {8};\n            \\end{axis}\n        \\end{tikzpicture}\n        \\caption{Right triangular mesh}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{0.4\\textwidth}\n        \\centering\n        \\begin{tikzpicture}\n            \\begin{axis}[\n                hide axis,\n                xmin=-0.3,\n                xmax=0.8,\n                axis equal,\n                width=\\textwidth,\n                height=\\textwidth,\n                no markers,\n                ticks=none,\n            ]\n            \\addplot[black] table {etri.dat};\n            \\node at (axis cs:0,  0.231) {1};\n            \\node at (axis cs:0.2,  0.115) {2};\n            \\node at (axis cs:0.4,  0.231) {3};\n            \\node at (axis cs:0.6,  0.1155) {4};\n            \n            \\node at (axis cs:0,  0.461) {5};\n            \\node at (axis cs:0.2,  0.577) {6};\n            \\node at (axis cs:0.4,  0.461) {7};\n            \\node at (axis cs:0.6,  0.577) {8};\n            \\end{axis}\n        \\end{tikzpicture}\n        \\caption{Equilateral triangular mesh}\n    \\end{subfigure}\n    \\hspace*{\\fill}%\n    \n    \\caption{Illustration of the natural ordering of mesh elements.}\n    \\label{fig:element-ordering}\n\\end{figure}\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n    \\hline\n    Hexagons &\n        \\highlightcell8 & 11 & \\highlightcell16 &\n        10 & 13 & 20 &\n        11 & 15 & 23 &\n        10 & 13 & 22 \\\\\n    Squares &\n        \\highlightcell8 & \\highlightcell10 & \\highlightcell16 &\n        \\highlightcell8 & \\highlightcell11 & \\highlightcell19 &\n        \\highlightcell7 & \\highlightcell10 & \\highlightcell17 & \n        \\highlightcell8 & \\highlightcell10 & \\highlightcell18 \\\\\n    Right triangles &\n        13 & 19 & 32 &\n        10 & 14 & 28 &\n        10 & 15 & 27 &\n        11 & 14 & 28\\\\\n    Equilateral triangles &\n        11 & 15 & 27 &\n        10 & 12 & 22 & \n        9 & 12 & 22 &\n        9 & 12 & 22 \\\\\n    \\end{tabular}\n    \\vspace{12pt}\n    \\caption{Iterations required for the GMRES iterative method with ILU(0) preconditioner to converge. \n    The smallest number of iterations in each column is highlighted.}\n    \\label{tab:gmres-ilu}\n\\end{table}\n\n\n\\subsection{Compressible Euler equations}\nThe compressible Euler equations of gas dynamics in two dimensions (see \\textit{e.g.} \\cite{Hartmann}) \nare given by \n\\begin{equation}\n    \\bm{u}_t + \\nabla\\cdot\\bm{f}(\\bm{u}) = 0,\n\\end{equation}\nfor\n\\begin{equation}\n    \\label{eq:euler-flux}\n    \\def1{1}\n    \\bm{u} = \\left(\\begin{array}{c} \\rho \\\\ \\rho u \\\\ \\rho v \\\\ \\rho E\\end{array}\\right),\n    \\qquad\n    \\bm{f}_1(\\bm{u}) = \\left(\\begin{array}{c} \\rho u \\\\ \\rho u^2 + p \\\\ \\rho uv \\\\ \\rho Hu\\end{array}\\right),\n    \\qquad\n    \\bm{f}_2(\\bm{u}) = \\left(\\begin{array}{c} \\rho v \\\\ \\rho uv \\\\ \\rho v^2 + p \\\\ \\rho Hv\\end{array}\\right),\n\\end{equation}\nwhere $\\rho$ is the density, $\\bm{v} = (u, v)$ is the fluid velocity, $p$ is the pressure, and $E$ is the specific \nenergy. The total enthalpy $H$ is given by \n\\begin{equation}\n    H = E + \\frac{p}{\\rho},\n\\end{equation}\nand the pressure is determined by the equation of state\n\\begin{equation}\n    p = (\\gamma - 1)\\rho \\left(E - \\frac{1}{2}\\bm{v}^2\\right),\n\\end{equation}\nwhere $\\gamma = c_p\/c_v$ is the ratio of specific heat capacities at constant pressure and constant volume.\n\nWe consider the model problem of an unsteady compressible vortex in a rectangular domain \\cite{Wang2013}. \nThe domain is taken to be a $20\\times15$ rectangle and the vortex is initially centered at $(x_0, y_0) = (5, 5)$. \nThe vortex is moving with the free-stream at an angle of $\\theta$. The exact solution is given by\n\\begin{gather}\n    u = u_\\infty \\left( \\cos(\\theta) - \\frac{\\epsilon ((y-y_0) - \\overline{v} t)}{2\\pi r_c} \n        \\exp\\left( \\frac{f(x,y,t)}{2} \\right) \\right),\\\\\n    u = u_\\infty \\left( \\sin(\\theta) - \\frac{\\epsilon ((x-x_0) - \\overline{u} t)}{2\\pi r_c} \n        \\exp\\left( \\frac{f(x,y,t)}{2} \\right) \\right),\\\\\n    \\rho = \\rho_\\infty \\left( 1 - \\frac{\\epsilon^2 (\\gamma - 1)M^2_\\infty}{8\\pi^2} \\exp((f(x,y,t))\n        \\right)^{\\frac{1}{\\gamma-1}}, \\\\\n    p = p_\\infty \\left( 1 - \\frac{\\epsilon^2 (\\gamma - 1)M^2_\\infty}{8\\pi^2} \\exp((f(x,y,t))\n        \\right)^{\\frac{\\gamma}{\\gamma-1}},\n\\end{gather}\nwhere $f(x,y,t) = (1 - ((x-x_0) - \\overline{u}t)^2 - ((y-y_0) - \\overline{v}t)^2)\/r_c^2$, \n$M_\\infty$ is the Mach number, $u_\\infty, \\rho_\\infty,$ and $p_\\infty$ are the \nfree-stream velocity, density, and pressure, respectively. The free-stream velocity is \ngiven by $(\\overline{u}, \\overline{v}) = u_\\infty (\\cos(\\theta), \\sin(\\theta))$. The \nstrength of the vortex is given by $\\epsilon$, and its size is $r_c$. \nWe choose the parameters to be $\\gamma = 1.4$, $M_\\infty = 0.5$, $u_\\infty = 1$, $\\theta = \\arctan(1\/2)$, \n$\\epsilon = 0.3$, and $r_c = 1.5$. \n\nIn the discontinuous Galerkin discretization of the Euler equations we use the Lax-Friedrichs numerical \nflux defined by\n\\begin{equation}\n    \\widehat{\\bm{F}}(\\bm{u}^+, \\bm{u}^-, \\bm{n}) = \\tfrac{1}{2} \\left(\n        \\bm{f}(\\bm{u}^-)\\cdot \\bm{n} + \\bm{f}(\\bm{u}^+)\\cdot \\bm{n} + \\alpha(\\bm{u}^- - \\bm{u}^+)\\right),\n\\end{equation}\nwhere $\\alpha$ is the maximum absolute eigenvalue over $\\bm{u}^-$ and $\\bm{u}^+$ of the matrix $B(\\bm{u}, \\bm{n})$ \ndefined by\n\\begin{equation}\n    B(\\bm{u}, \\bm{n}) = J_{\\bm{f}_1} n_1 + J_{\\bm{f}_2} n_2,\n\\end{equation}\nwhere $J_{\\bm{f}_1}$ and $J_{\\bm{f}_2}$ are the Jacobian matrices of the components of the numerical \nflux function $\\bm{f}$ defined in equation \\eqref{eq:euler-flux}.\n\nWe use the backward Euler time discretization, but remark that\n\\eqref{eq:semi-disc-dg} results in a nonlinear set of equations, which are\nsolved using Newton's method. Each iteration of Newton's method requires solving\na linear equation of the form \\eqref{eq:be}. We set $h = 1$, and consider three\ntime steps, $k_1 = 0.03h$, $k_2 = 2 k_1$, $k_3 = 4 k_1$. We use piecewise\npolynomials of degrees $p = 0, 1, 2, 3$. Each Newton solve requires between 3 to\n8 iterations to converge to within a tolerance of \n$5 \\times 10^{-13}$. The tolerance used for the linear solvers is the same as in \nthe previous test cases.\n\n\\subsubsection{The Block Jacobi method}\nEach iteration of Newton's method requires the solution of a linear system of \nequations. We solve these systems using the block Jacobi method. We compute\nthe  total the number of Jacobi iterations required to complete one solve of \nNewton's method, and report the results in Table \\ref{tab:euler-jac}. We note \nthat for each choice of $p$ and time step $k$, the hexagonal mesh required the \nfewest number of block Jacobi iterations. As in the previous numerical \nexperiments, we do not see a decrease in performance for the hexagonal elements \nin the case of $p=3$. The square mesh resulted in the second-smallest number of \niterations for most of the cases considered, while the two configurations of \ntriangles resulted in generally similar numbers of iterations.\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n            \\hline\n            Hexagons                & \n        \\highlightcell32 & \\highlightcell49 & \\highlightcell78 & \n        \\highlightcell31 & \\highlightcell50 & \\highlightcell83 &\n        \\highlightcell50 & \\highlightcell90 & \\highlightcell158 & \n        \\highlightcell53 & \\highlightcell97 & \\highlightcell171 \\\\\n            Squares                 & \n        34 & 51 & 89 &\n        \\highlightcell31 & 54 & 92 &\n        54 & 99 & 181 &\n        55 & 105 & 201  \\\\\n            Right triangles         & \n        37 & 56 & 97 &\n        41 & 64 & 112 &\n        58 & 101 & 189 &\n        59 & 113 & 217 \\\\\n            Equilateral triangles   & \n        37 & 57 & 95 &\n        39 & 62 & 113 &\n        54 & 99 & 179 &\n        60 & 114 & 215\n        \\end{tabular}\n        \\vspace{8pt}\n    \\caption{Block Jacobi iterations required per Newton solve of the \n    compressible Euler equations. The lowest number of iterations in each \n    column is highlighted.}\n    \\label{tab:euler-jac}\n\\end{table}\n\n\\subsubsection{The GMRES method}\nWe now repeat the above test case, using the GMRES method to solve the resulting\nlinear systems. We consider both the block Jacobi and block ILU(0)\npreconditioners. We then compute the total number of GMRES iterations required\nto complete one solve of Newton's method. As in Section \\ref{sec:adv-gmres}, the\nordering of the mesh elements has a significant effect on the effectiveness of\nthe block ILU(0) approximate factorization. For this reason, we use the natural\nordering of elements, depicted in Figure \\ref{fig:element-ordering}. We present\nthe results for the block Jacobi preconditioner in Table\n\\ref{tab:euler-gmres-jac}, and for the block ILU(0) preconditioner in Table\n\\ref{tab:euler-gmres-ilu}. With the block Jacobi preconditioner, the hexagonal\nmesh required the smallest number of iterations for all test cases considered,\nand the square mesh the second-smallest. In the case of the block ILU(0)\npreconditioner, the square mesh required the fewest number of iterations, with\nthe hexagonal mesh usually requiring the second-smallest number of iterations.\nAs we observed in Section \\ref{sec:adv-gmres}, the number of iterations required\nfor both the block Jacobi method and GMRES with the block Jacobi preconditioner\nscales quite poorly with increasing timesteps. The number of GMRES iterations\nrequired when using the block ILU(0) preconditioner is significantly better.\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n            \\hline\n            Hexagons                & \n        \\highlightcell55 & \\highlightcell74 & \\highlightcell106 & \n        \\highlightcell50 & \\highlightcell92 & \\highlightcell126 &\n        \\highlightcell61 & \\highlightcell110 & \\highlightcell153 & \n        \\highlightcell76 & \\highlightcell141 & \\highlightcell195 \\\\\n            Squares                 & \n        62 & 84 & 155 &\n        52 & 93 & 132 &\n        67 & 126 & 185 &\n        78 & 149 & 222 \\\\\n            Right triangles         & \n        63 & 87 & 162 &\n        81 & 106 & 184 &\n        96 & 132 & 242 &\n        85 & 159 & 299\\\\\n            Equilateral triangles   & \n        66 & 90 & 167 &\n        81 & 108 & 187 &\n        72 & 133 & 197 &\n        85 & 161 & 245\n        \\end{tabular}\n        \\vspace{8pt}\n    \\caption{GMRES with block Jacobi preconditioner. Iterations required per \n    Newton solve of the compressible Euler equations. The lowest number of \n    iterations in each column is highlighted.}\n    \\label{tab:euler-gmres-jac}\n\\end{table}\n\n\\begin{table}[h]\n    \\centering\n    \\tiny\n    %\n    \\setlength{\\tabcolsep}{8pt}\n    \\begin{tabular}{r | lll | lll | lll | lll}\n    & \\multicolumn{3}{c|}{$p=0$} & \\multicolumn{3}{c|}{$p=1$} & \\multicolumn{3}{c|}{$p=2$}\n    & \\multicolumn{3}{c}{$p=3$}\\\\\n   \n    & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$ & $k_1$ & $k_2$ & $k_3$\\\\\n    \\hline\n            Hexagons                & \n        \\highlightcell24 & 32 & \\highlightcell42 & \n        \\highlightcell21 & 36 & 48  &\n        29 & 48 & 57 &\n        29 & 50 & 64 \\\\\n            Squares                 & \n        \\highlightcell24 & \\highlightcell28 & 45 &\n        \\highlightcell21 & \\highlightcell33 & \\highlightcell40 &\n        \\highlightcell24 & \\highlightcell41 & \\highlightcell49 &\n        \\highlightcell27 & \\highlightcell48 & \\highlightcell60 \\\\\n            Right triangles         & \n        31 & 40 & 70 &\n        35 & 40 & 60 &\n        36 & 48 & 69 &\n        31 & 49 & 75 \\\\\n            Equilateral triangles   & \n        28 & 37 & 65 &\n        37 & 44 & 70 &\n        33 & 56 & 68 &\n        38 & 64 & 80  \\\\\n        \\end{tabular}\n        \\vspace{8pt}\n    \\caption{GMRES with block ILU(0) preconditioner. Iterations required per \n    Newton solve of the compressible Euler equations. The lowest number of \n    iterations in each column is highlighted.}\n    \\label{tab:euler-gmres-ilu}\n\\end{table}\n\n\\subsection{Inviscid flow problems}\n\nThe following two numerical experiments extend the above results to\nlarger-scale, more realistic flow problems. These problems, in contrast to the\npreceding test cases, are characterized by a large number of degrees of freedom,\nthe presence of geometric features and wall boundary conditions, variably-sized\nmesh elements, and shocks. As in the previous section, the equations considered\nhere are the compressible Euler equations. For the following two problems, we\nchoose the finite element function space to consist of piecewise constant\nfunctions (corresponding to $p=0$), which results in a finite-volume-type\ndiscretization. This choice of discretization allows for the solution of\nproblems with shocks, without the use of slope limiters, artificial viscosity,\nor other shock-capturing techniques \\cite{leveque2002finite}. The Roe numerical\nflux is used as an approximate Riemann solver for these problems.\n\n\\subsubsection{Subsonic flow over a circular cylinder}\n\nFor a first test case, we consider the inviscid flow over a circular cylinder at\nMach 0.2. The computational domain is defined as $\\Omega = R \\setminus C$, where\n$R = [-10, 30] \\times [-10, 20]$, and $C$ is a disk of radius 1 centered at the\npoint $(5,5)$. Farfield boundary conditions are enforced on $\\partial R$, and a\nno normal flow condition is enforced on $\\partial C$. The freestream velocity is\ntaken to be unity in the $x$-direction, and $\\rho_\\infty = 1$. For this test\ncase we use four unstructured meshes, two consisting entirely of triangles, and\ntwo consisting of mixed polygons, generated using the PolyMesher algorithm\n\\cite{Talischi:2012}. All the meshes are created using a gradient-limited\nelement size function that determines the initial distribution of seed points\naccording to the rejection method \\cite{persson2005mesh}, such that the element\nedge length near the surface of the cylinder is about one-fifth the edge length\nof elements away from the cylinder. For both the triangular and polygonal\nmeshes, we consider a coarse mesh, with 15{,}404 elements, and a fine mesh with\n62{,}270 elements. Thus, the average area of each element is the same for both\nthe polygonal and triangular meshes. Additionally, the number of degrees of\nfreedom in the solution is the same, allowing for a fair comparison. The coarse\npolygonal mesh, and a zoom-in around the surface of the cylinder are shown in\nFigure \\ref{fig:cyl-mesh}.\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\begin{subfigure}{3.35in}\n        \\centering\n        \\includegraphics[width=3.3in]{CylMesh}\n       \n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{2.55in}\n        \\centering\n        \\includegraphics[width=2.5in]{CylMeshZoom}\n       \n    \\end{subfigure}%\n   \n    \n    \\caption{Overview of the coarse mesh with 15{,}404 elements, with zoom-in \n             showing polygonal elements near the surface of the cylinder.}\n    \\label{fig:cyl-mesh}\n\\end{figure}\n\nStarting from freestream initial conditions, we integrate the equations until $t\n= 5 \\times 10^{-3}$ in order to obtain a representative solution. Using this\nsolution, we then compute 10 time steps using a third-order $A$-stable DIRK\nmethod \\cite{Alexander1977}. Each stage of the DIRK method requires the solution\nof a nonlinear system of equations, which we solve by means of Newton's method.\nIn each iteration of Newton's method, we solve the resulting linear system of\nthe form \\eqref{eq:be} using both the block Jacobi method and the preconditioned\nGMRES method.\nThe nonlinear system is solved to within a tolerance of \n$10^{-8}$, and each linear system is solved using a relative tolerance of $10^{-5}$.\n For the GMRES method, we consider two preconditioners: block\nJacobi, and block ILU(0). In order to compare the iterative solver performance\ndifferences between meshes, we compute the total number of solver iterations\nrequired to complete all 10 time steps. The results for the GMRES method are \nshown in Table \\ref{tab:cylinder-gmres}, and for the block Jacobi solver in\nTable \\ref{tab:cylinder-jacobi}.\n\n\\begin{table}[t] \n\\caption{Total GMRES iterations per 10 time steps for inviscid flow\n         over a circular cylinder.}\n\\label{tab:cylinder-gmres}\n\\centering\n\\begin{subtable}{0.99\\linewidth}\n\\centering\n\\caption{Coarse grid with 15{,}404 elements}\n\\begin{tabular}{lcccc|cc}\n\\toprule\n & \\multicolumn{2}{c}{ILU} & \\multicolumn{2}{c|}{Jacobi} & \n  \\multicolumn{2}{c}{Ratios} \\\\\n$\\Delta t$ & Polygonal & Triangular & Polygonal & Triangular & ILU & Jacobi \\\\\n\\midrule\n$1.0 \\times 10^{-1}$ & 793 & 932 & 2092 & 3126 & 0.85 & 0.67 \\\\\n$2.5 \\times 10^{-1}$ & 1569 & 1829 & 4405 & 6870 & 0.86 & 0.64 \\\\\n$5.0 \\times 10^{-1}$ & 2470 & 3090 & 7145 & 11859 & 0.80 & 0.60 \\\\\n$1.0$                & 3651 & 4486 & 11054 & 18880 & 0.81 & 0.59 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{subtable} \\\\\n\\begin{subtable}{0.99\\linewidth}\n\\centering\n\\caption{Fine mesh with 95{,}932 elements}\n\\begin{tabular}{lcccc|cc}\n\\toprule\n & \\multicolumn{2}{c}{ILU} & \\multicolumn{2}{c|}{Jacobi} & \n  \\multicolumn{2}{c}{Ratios} \\\\\n$\\Delta t$ & Polygonal & Triangular & Polygonal & Triangular & ILU & Jacobi \\\\\n\\midrule\n$1.0 \\times 10^{-1}$ & 1443 & 1673 & 4075 & 6137 & 0.86 & 0.66 \\\\\n$2.5 \\times 10^{-1}$ & 2998 & 3344 & 8732 & 12741 & 0.90 & 0.69 \\\\\n$5.0 \\times 10^{-1}$ & 4720 & 5423 & 14084 & 21882 & 0.87 & 0.64 \\\\\n$1.0$                & 7205 & 8151 & 22814 & 34706 & 0.88 & 0.66 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{subtable}\n\\end{table}\n\\begin{table}[t] \n\\caption{Total block Jacobi iterations per 10 time steps for inviscid \n         flow over a circular cylinder.}\n\\label{tab:cylinder-jacobi}\n\\centering\n\\begin{subtable}{0.49\\linewidth}\n\\centering\n\\caption{Coarse grid with 15{,}404 elements}\n\\begin{tabular}{llll}\n\\toprule\n$\\Delta t$ & Polygonal & Triangular & Ratio \\\\\n\\midrule\n$1.0 \\times 10^{-1}$ & 2474 & 3159 & 0.78 \\\\\n$2.5 \\times 10^{-1}$ & 4895 & 6697 & 0.73 \\\\\n$5.0 \\times 10^{-1}$ & 7882 & 12158 & 0.65 \\\\\n$1.0$              & 13181 & 19072 & 0.69 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{subtable}\n\\begin{subtable}{0.49\\linewidth}\n\\centering\n\\caption{Fine mesh with 95{,}932 elements}\n\\begin{tabular}{llll}\n\\toprule\n$\\Delta t$ & Polygonal & Triangular & Ratio \\\\\n\\midrule\n$1.0 \\times 10^{-1}$ & 4788 & 6281 & 0.76 \\\\\n$2.5 \\times 10^{-1}$ & 9609 & 12406 & 0.77 \\\\\n$5.0 \\times 10^{-1}$ & 15580 & 20946 & 0.74 \\\\\n$1.0$                & 26628 & 33934 & 0.78 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{subtable}\n\\end{table}\n\nThese results demonstrate a consistent trend, corroborating both the numerical\nresults and the analysis from the previous sections. When using the block Jacobi\nsolver or GMRES with block Jacobi preconditioner, the polygonal mesh results in\nconvergence in between 60--70\\% of the iterations required for the triangular\nmesh. The effect is smaller when using the ILU(0) preconditioner, but we do\nstill observe a modest reduction in the number of iterations required. When \nusing the block Jacobi iterative solver, we observe iteration counts very \nsimilar to when using GMRES with block Jacobi as a preconditioner. In these \ncases, the polygonal mesh requires between 70--80\\% of the iterations as the \nall-triangular mesh.\n\n\\subsubsection{Supersonic flow over a circular cylinder}\n\nThe next numerical example is designed to investigate the performance of the\niterative solvers for steady-state problems, in the presence of shocks and\n$h$-adapted meshes. For this problem, we let the domain be $\\Omega = R \\setminus\nC$, where $ R =[0,5]\\times[0,10]$, and, as before, $C$ is a circle of radius one\ncentered at $(5,5)$. Freestream conditions are enforced at the left, top, and\nbottom boundaries, an inviscid wall condition is enforced on the boundary of the\ncylinder, and an outflow condition is enforced on the right boundary. The Mach\nnumber is set to $M = 2.0$, resulting in the formation of a shock upstream from\nthe cylinder. In order to accurately capture the shock, we refine the mesh in\nits vicinity. As in the previous case, we consider a set of four meshes, two\nall-triangular, and two polygonal. For both the triangular and polygonal meshes,\nwe consider coarse and fine versions, with 31{,}162 and 95{,}932 elements,\nrespectively. The coarse mesh is depicted in Figure \\ref{fig:supersonic-mesh},\nwith Mach isolines overlaid to indicate the position of the shock. Additionally,\nMach contours of the steady-state solution are shown in Figure\n\\ref{fig:supersonic-contours}.\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\begin{subfigure}{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=1.7in]{SupersonicMesh}\n        \\caption{Coarse mesh for supersonic test problem, showing Mach isolines\n                 for steady-state solution}\n        \\label{fig:supersonic-mesh}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=2.3in]{SupersonicContours}\n        \\caption{Contours of Mach number for steady state solution}\n        \\label{fig:supersonic-contours}\n    \\end{subfigure}%\n   \n    \\caption{Overview of coarse polygonal mesh with 31{,}162 elements, showing \n             Mach number contours for steady-state solution.}\n    \\label{fig:supersonic}\n\\end{figure}\n\nBeginning with freestream initial conditions, the solution rapidly approaches a\nsteady state. We integrate in time until $t = 100$ in order to obtain an\nsolution which can be used as an initial guess for the steady-state Newton\nsolve. Then, starting with this solution, we set the time-derivative of the\nsolution to zero and solve the resulting nonlinear equations using Newton's\nmethod to find a steady-state solution. The resulting linear system that is\nrequired to be solved at each iteration can be thought of as corresponding to\nequation \\eqref{eq:be}, where formally we set $k = \\infty$.\nThe nonlinear system is solved to within a tolerance of \n$10^{-10}$, and each linear system is solved using a relative tolerance of $10^{-5}$.\nSince the mass\nmatrix in \\eqref{eq:be} acts to regularize the linear system, the conditioning\nbecomes worse for larger values of $k$, and the number of iterations required\nper linear solve grows. Hence, effective preconditioners are particularly\nimportant for the solution of such steady-state problems. For these problems, \nthe block Jacobi iterative solver did not converge in fewer than 10{,}000 \niterations, and so we consider only the GMRES method, using block ILU(0) and \nblock Jacobi preconditioners.\n\nWe present the comparison of iteration counts for this problem in Table \n\\ref{tab:supersonic-results}. On the coarse meshes, the ILU(0) preconditioner \nrequired about 73\\% as many iterations on the polygonal mesh when compared with\nthe triangular mesh. This difference is more significant when using the block \nJacobi preconditioner, consistent with the results observed in previous section.\nIn this case, the polygonal mesh requires only slightly more than one third the \nnumber of iterations as the all-triangular mesh. On the fine mesh, there are \nclose to half a million degrees of freedom. For a problem of this scale, we did \nnot observe convergence in less than 10{,}000 iterations per linear solve using \nthe block Jacobi preconditioner, and so we only compare performance using the \nblock ILU(0) preconditioner. In this case, the polygonal mesh required about \nhalf as many iterations per steady-state solve when compared with the \nall-triangular mesh.\n\n\\begin{table}[h!] \n\\caption{Total GMRES iterations per steady-state solve for supersonic \n         flow over a cylinder.}\n\\label{tab:supersonic-results}\n\\centering\n\\begin{subtable}{0.49\\linewidth}\n\\caption{Coarse grid with 31{,}162 elements}\n\\begin{tabular}{lccc}\n\\toprule\n & Polygonal & Triangular & Ratio \\\\\n\\midrule\nILU & 469 & 640 & 0.73 \\\\\nJacobi & 2340 & 6464 & 0.36 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{subtable}\n\\begin{subtable}{0.49\\linewidth}\n\\caption{Fine mesh with 95{,}932 elements}\n\\begin{tabular}{lccc}\n\\toprule\n & Polygonal & Triangular & Ratio \\\\\n\\midrule\nILU & 953 & 1947 & 0.49 \\\\\nJacobi & -- & -- & -- \\\\\n\\bottomrule\n\\end{tabular}\n\\end{subtable}\n\\end{table}\n}\n\n\\section{Conclusions}\\label{sec:conclusions}\nIn this paper we have analyzed the effect of the generating pattern of a regular\nmesh on the convergence of iterative linear solvers applied to implicit\ndiscontinuous Galerkin discretizations. We considered four generating patters: a\nhexagon, a square, two right triangles, and two equilateral triangles.\n\nA classical von Neumann analysis applied to the constant-velocity advection\nequation allowed us to compute the eigenvalues of the block Jacobi matrix, and\ntherefore estimate the speed of convergence of the block Jacobi method. In more\nthan half of the cases considered, the hexagonal generating pattern resulted in\nthe smallest eigenvalues, and in the remaining cases, the square generating\npattern resulted in the smallest eigenvalues.\n\nIn order to extend these results beyond the case of the constant-velocity\nadvection equation, we performed numerical experiments on the variable-velocity\nadvection equation and compressible Euler equations. In the case of the\nadvection equation, in all but one case the hexagonal mesh resulted in the\nfastest convergence, and in the remaining case the square mesh resulted in the\nfastest convergence. In the case of the Euler equations, the hexagonal mesh\nresulted in the fastest convergence in all test cases.\n\nWe additionally considered two irregular meshes resulting from the random\nperturbation of a set of regularly-spaced generating points. We obtain a\ntriangular mesh by performing the Delaunay triangulation on these points, and we\nobtain a polygonal mesh by constructing the Voronoi diagram dual to the Delaunay\ntriangulation. Solving the advection equation on these irregular meshes, we\nobserved that the block Jacobi method converged faster on the polygonal mesh in\nevery test case. Additionally, we performed numerical experiments examining the\nperformance of the GMRES iterative method when used with the ILU(0)\npreconditioner. We found that in all of the test cases, the square generating\npattern resulted in the fewest number of GMRES iterations, and in all but two\ncases, the hexagonal generating pattern resulted in the second-fewest number of\niterations.\n\nFor a final set of numerical experiments, we performed two inviscid\nfluid flow simulations on sets of coarse and fine meshes. Each mesh was either\nall-triangular, or was composed of arbitrary polygons. We measured iteration\ncounts for both time-dependent and steady-state problems, using the block Jacobi\nmethod, and GMRES with block ILU(0) and block Jacobi preconditioners. We found\nthat the polygonal meshes resulted in faster convergence of the iterative\nsolvers, with a larger difference being observed for the block Jacobi method and\npreconditioner. This difference was more pronounced for the steady-state\nproblem, with a quite significant difference observed on the fine mesh using\nGMRES with ILU(0).\n\nThese results suggest that certain types of polygonal meshes have the advantage\nof rapid convergence of iterative solvers. Future research directions involve\nthe study of accuracy of DG methods on polygonal and polyhedral meshes,\nefficient computation of quadrature rules over arbitrary polygonal domains and\nthe extension of the above results to three spatial dimensions.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPhysics community around the world has been tirelessly working for the\npast months to find different ways of increasing the superconducting\ntransition temperature $T_{c}$ following the discovery of\nsuperconductivity at 26 K in the rare-earth based ($RE$OFeAs) system\nLaO$_{1-x}$F$_{x}$FeAs ($x$ = 0.05 - 0.12) \\cite{Kamihara08a}.  Spirited\nsearch by the experimentalists has led to eventually raising $T_{c}$\nto 55 K for another member of this family of compounds,\nSmFeAsO$_{0.9}$F$_{0.1}$ \\cite{Ren08a}. Shortly afterwards, another\nfamily of Fe-based compounds $A$Fe$_{2}$As$_{2}$ ($A$ = Ca, Sr, Ba,\nEu) was also found to be superconducting upon hole doping\n\\cite{definition} on the $A$ site with a maximum $T_{c}$ = 38 K\n\\cite{Sasmal08a,GFChen08a,Rotter08b}.  These discoveries are followed\nby the announcements of other new parent compounds: LiFeAs, FeSe,\nSrFeAsF with a maximum $T_{c}$ of 18 K \\cite{WangXC08a}, 14 K\n\\cite{WangXC08a} (27 K using pressure \\cite{Mizuguchi08a}) and 56 K\n\\cite{Wu08b}, respectively.  The basic features, that are common to\nmany of these new parent compounds, are the anti-ferromagnetic\nordering of the Fe-spins at $T_{N}$ $\\approx$ 100--200 K, and the\nquasi-2D nature of the electronic structure.  All the members\nbelonging to the above mentioned families have been shown to\nsuperconduct upon either ``hole'' or ``electron doping'' only with the\nexception of LiFeAs, wherein it is considered that the Li layer acts\nas a charge reservoir for the system \\cite{WangXC08a}.\nSuperconductivity also emerges upon application of hydrostatic\npressure for certain members of the $RE$OFeAs, $A$Fe$_{2}$As$_{2}$ and\nFeSe families.  Notwithstanding the abundant research that has already\nbeen performed, the microscopic nature of the mechanism of\nsuperconductivity has thus far remained elusive.  A systematic study\nof the many members within one particular family, using both\nexperimental and theoretical techniques, would pave way to a more\nconcrete understanding of the electronic structure of the normal\nstate. Although the $RE$OFeAs systems have larger $T_{c}$ values,\ntheir sample preparation, characterization and quality, as is the case\nfor many oxides, have to be considered with great care. Alternatively\nit has been demonstrated that the $A$Fe$_{2}$As$_{2}$ systems could be\nsynthesized comparatively easier.  Consequently, here we carry out a\nsystematic investigation using both theoretical and experimental\ntechniques for the $A$Fe$_{2}$As$_{2}$ systems.\n\nAll the $A$Fe$_{2}$As$_{2}$ compounds crystallize in the tetragonal\nThCr$_{2}$Si$_{2}$-type structure at room temperature.  They all\nexhibit a structural transition upon cooling to an orthorhombic\nlattice ($T_{0}$ for Ca $\\approx$ 171 K \\cite{Ronning08a}, Sr\n$\\approx$ 205 K \\cite{Krellner08a}, Ba $\\approx$ 140 K\n\\cite{Rotter08a}, Eu $\\approx$ 200 K \\cite{Jeevan08a}).  The\nstructural transition is coupled with an antiferromagnetic ordering of\nthe Fe moments with a wave vector $Q$ = [1,0,1] for the\nspin-density-wave (SDW) pattern. Suitable substitution on either the\n$A$ site or the Fe site can suppress the magnetic ordering, and then\nthe system becomes superconducting for certain ranges of doping (for\nexample, maximum $T_{c}$ = ~38 K (Ba,K)Fe$_{2}$As$_{2}$\n\\cite{Rotter08b}, ~32 K (Eu,K)Fe$_{2}$As$_{2}$ \\cite{Jeevan08b}, ~21 K\nSr(Fe,Co)$_{2}$As$_{2}$ \\cite{LeitheJasper08b},\nBa(Fe,Co)$_{2}$As$_{2}$ \\cite{Sefat08b}).  Superconductivity can also\nbe induced in ``undoped'' and ``under-doped'' compounds by applying\npressure \\cite{Kumar08a}.\n\nIn the following sections we describe the microscopic picture of the\nmagneto-structural transition and the effects of external pressure,\nchemical pressure and charge doping on the $A$Fe$_{2}$As$_{2}$\nsystems.  In order to make the manuscript more easily readable, we\nintroduce the following abbreviations: $RE$OFeAs: $1111$,\n$A$Fe$_{2}$As$_{2}$: $122$, LiFeAs: $111$, FeSe: $11$, SrFeAsF:\n$1111^{\\prime}$; ferromagnetic: FM, checkerboard (nearest neighbour)\nantiferromagnetic: NN-AFM, columnar\/stripe-type antiferromagnetic\norder of the Fe-spins: SDW (spin-density-wave).\n\n\\section{Methods}\n\n\\subsection{{\\bf Theory}}\n\nWe have performed density functional band structure calculations using\na full potential all-electron local orbital code FPLO\n\\cite{fplo1,fplo2,version} within the local (spin) density\napproximation (L(S)DA) including spin-orbit coupling when needed.  The\nPerdew-Wang \\cite{perdew} parametrization of the exchange-correlation\npotential is employed.  Density of states (DOS) and band structures\nwere calculated after converging the total energy on a dense $k$-mesh\nwith 24$\\times$24$\\times$24 points.  The strong Coulomb repulsion in\nthe Eu 4$f$ orbitals are treated on a mean field level using the\nLSDA+$U$ approximation in the atomic-limit double counting scheme\n\\cite{czyzyk}.  Results we present below use the LSDA+$U$ method\n\\cite{aza} in the rotationally invariant form \\cite{laz}.  In\naccordance with the widespread belief that in the new Fe-based\nsuperconducting compounds the Fe $3d$ electrons have a more itinerant\ncharacter than a localized one, and thereby are much less correlated\nin comparison to the Cu $3d$ electrons in the high-$T_c$ cuprates, we\ndid not apply the LSDA+$U$ approximation to the Fe 3$d$ states.\nEffects of doping on either the cation site or the Fe site were\nstudied using the virtual crystal approximation (VCA) treatment.  The\nresults obtained via VCA were cross checked using supercells for\ncertain doping concentrations.  The crystal structures are optimized\nat different levels to investigate or isolate effects that may depend\nsensitively to certain structural features.  The full relaxation of\nthe unit cell of the 122 systems at low temperatures involves\noptimizing $a\/b$ and $c\/a$ ratios in addition to relaxing the As-$z$\nposition. \n\n\\subsection{{\\bf Experimental}}\\label{experimental}\n\nPolycrystalline samples were prepared by sintering in glassy-carbon\ncrucibles which were welded into tantalum containers and sealed into\nevacuated quartz tubes for heat treatment at 900\\textdegree C for 16\nhours with two regrinding and compaction steps. First precursors SrAs,\nCo$_{2}$As, Fe$_{2}$As, Mn$_{2}$As and NiAs were synthesized from\nelemental powders sintered at 600\\textdegree C for 48 h (Mn, Fe, Co,\nNi 99.9 wt.\\%; As 99.999 wt.\\%; Sr 99.99 wt.\\%). These educts were\npowdered, blended in stoichiometric ratios, compacted to pellets, and\nheat treated. All sample handling was done inside argon-filled glove\nboxes.  Crystals were grown in glassy-carbon crucibles by a modified\nself flux method \\cite{Morinaga2008,Sefat08b} in melts with\ncompositions SrFe$_{5-x}$Co$_{x}$As$_{5}$ (0.5 $\\leq$ $x$ $\\leq$ 0.85)\nby cooling from 1250\\textdegree C to 1100\\textdegree C within 48\nhours. The melt was spun off at 1100\\textdegree C using a centrifuge\n\\cite{Bostrom2006}. Metallographic investigations were performed on\npolished surfaces of selected secured crystal\nplatelets. Electron-probe microanalysis (EPMA) with\nwavelength-dispersive analysis was accomplished in a Cameca SX100\nmachine.  Crystal X1 was grown in a flux\nSrFe$_{4.25}$Co$_{0.75}$As$_{5}$ and crystal X2 in a flux\nSrFe$_{0.15}$Co$_{0.85}$As$_{5}$.  For crystal X1 from EPMA the\ncomposition (in at.\\%)\nSr$_{19.9(2)}$Fe$_{36.2(1)}$Co$_{4.2(1)}$As$_{39.7(1)}$ was found\nwhich corresponds to SrFe$_{2-x}$Co$_{x}$As$_{2}$ with $x$ $\\approx$\n0.21.  For crystal X2 from EPMA the composition (in at.\\%)\nSr$_{19.5(2)}$Fe$_{35.5(1)}$Co$_{5.0(2)}$As$_{39.9(2)}$ was found\nwhich corresponds to SrFe$_{2-x}$Co$_{x}$As$_{2}$ with $x$ $\\approx$\n0.25. Crystals grown from a flux SrFe$_{4.5}$Co$_{0.5}$As$_{5}$ had a\ncomposition of Sr$_{19.9(3)}$Fe$_{36.9(1)}$Co$_{3.2(1)}$As$_{39.9(2)}$\ncorresponding to SrFe$_{2-x}$Co$_{x}$As$_{2}$ with $x \\approx$ 0.15\nand showed no superconductivity. All crystals grown up to now exhibit\nsome inclusions of flux-material which can be seen in\nFig.~\\ref{crystal} (for crystal X2 the second phase has the\ncomposition Sr$_{3(1)}$Fe$_{44(1)}$Co$_{8(1)}$As$_{45(1)}$ (in\nat.\\%)).\n\\begin{figure}[h]\n  \\begin{center}\n    \\begin{minipage}[t]{0.46\\linewidth}\n      \\raisebox{-3cm}{\\includegraphics[clip=, width=1.2\\linewidth]{crystal.ps}}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.52\\linewidth}\n      \\caption{\\label{crystal} Micrograph of a polished crystal from the same \n       batch crystal X2 was selected. Light grey phase: bulk crystal. \n       Dark grey phase: inclusions of flux. Black regions: micro-cracks and\n       cavities. }\n    \\end{minipage}\n  \\end{center}\n\\end{figure}\n\n\\section{Results - Theory}\n\n\\subsection{{\\bf Ambient-temperature phase : tetragonal }}\n\nWe begin with comparing the DOS computed for a representative member\nof each family of the iron pnictide compounds. In these calculations,\nexperimental values of the ambient-temperature tetragonal lattice\nparameters and atomic positions were used for all the\nsystems. Collected in Fig.~\\ref{dos} are the non-magnetic total and\norbital-resolved DOS for five systems: $1111$, $122$, $111$, $11$,\n$1111^{\\prime}$. The states close to the Fermi energy ($E_{F}$) in all\nthese systems are comprised mainly of Fe $3d$ contributions. The\ncontribution of the pnictide atom (or Se in the case of FeSe) to the\nFermi surface is small but non-zero.  A pseudo-gap-like feature in the\nDOS slightly above the $E_{F}$ is common to all the systems.  At the\noutset, all the five systems look quite similar to one another, but\nslight differences are already visible when analyzing the Fe 3$d$\norbital resolved DOS, presented on right panels in Fig.~\\ref{dos}. The\ncontribution of the Fe 3$d_{x^{2}-y^{2}}$ orbital (the orbital\npointing directly towards the nearest neighbour Fe ions) is the\nlargest close to $E_{F}$ for all the systems.  The corresponding bands\n(not shown here) are highly dispersive in the $a-b$ plane and remain\nflat along the $c$-axis, indicative of the quasi-2D nature of this\nband.  The distance of the 3$d_{x^{2}-y^{2}}$ edge from the $E_{F}$\nvaries for the different systems and is the largest for the 111 family\nand smallest for the 1111 family. The second largest contribution to\nthe Fermi surface comes from the doubly degenerate 3$d_{xz}$ and\n3$d_{yz}$ orbitals and this feature is again consistent for all the\niron pnictide compounds.\n\n\\begin{figure}[H]\n  \\begin{center}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=LaOFeAs_dos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=LaOFeAs_Fedos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=SrFe2As2_dos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=SrFe2As2_Fedos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=LiFeAs_dos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=LiFeAs_Fedos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=FeSe_dos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=FeSe_Fedos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=SrFeAsF_dos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.38\\linewidth}\n      \\epsfig{file=SrFeAsF_Fedos.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n\\caption{\\label{dos}Comparison of total and site-resolved density of\n  states (DOS) per cell (left panel) and Fe 3$d$ orbital resolved DOS\n  (right panel) of a representative member for each of the new\n  superconducting family of compounds. LaOFeAs:$1111$,\n  SrFe$_{2}$As$_{2}$:$122$, LiFeAs:$111$, FeSe:$11$,\n  SrFeAsF:$1111^{\\prime}$. The solid vertical lines at zero energy\n  denote the Fermi level $E_{F}$.}\n  \\end{center}\n\\end{figure}\n\n\n\\subsection{{\\bf Structural distortion vs. magnetic order}}\n\n\\subsubsection{{\\bf Tetragonal to orthorhombic distortion}}\n\\label{str}\n\nAs discussed above, the FeAs-based compounds crystallizing in\ndifferent structures have very similar electronic properties. However,\nbetween the $1111$ and the $122$ families there is an important\ndifference in regard to structural and magnetic transitions. In the\nformer, the transition temperatures for the structural transition are\n10--20 K higher than those of the magnetic one. On the other hand, for\nthe $122$ family the structural and the magnetic transitions are found\nto be coupled and occur at the same temperature. By first-principles\ncalculations we explored the nature of this intimate connection\nbetween the two transitions for the $122$ systems.\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\begin{minipage}[t]{0.46\\linewidth}\n      \\raisebox{-8cm}{\\includegraphics[clip=, width=1.2\\linewidth]{afm_stripe_comp.eps}}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.52\\linewidth}\n      \\caption{\\label{dist} Total energy as a function of $b\/a$ for\n        the $A$Fe$_{2}$As$_{2}$ ($A$ = Ca, Sr, Eu, Ba) systems with\n        two possible spin arrangements between the Fe spins. In NN-AFM\n        the spins of all the nearest neighbours are anti-parallel to\n        each other. In the SDW pattern, the spins along the longer\n        $a$-axis are anti-parallel while the spins along the shorter\n        $b$-axis are parallel. The solid vertical lines denote the\n        minimum value of the energy. The minimum energy for each\n        system has been set to zero. For clarity, Sr122, Eu122 and\n        Ba122 curves are offset by 5, 10 and 15 meV.}\n    \\end{minipage}\n  \\end{center}\n\\end{figure}\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption{\\label{bovera}Comparison of $b\/a$ obtained from LSDA for the\n  SDW pattern with experimental reports. The references from which the\n  experimental numbers are obtained are indicated. With the exception\n  of Ca122, (more detailed discussion, see text) the trend in the\n  in-plane axis distortion for the orthorhombic structure is\n  consistent with experimental findings.}\n\\begin{tabular}{cccc}\n        & $b\/a$-LSDA     &  $b\/a$-expt    & Ref \\\\\n\\hline\nCa122   & 0.9809        &  0.9898       & \\cite{Kreyssig08a}\\\\\nSr122   & 0.9841\t &  0.9889       & \\cite{Jesche08a} \\\\\nEu122   & 0.9850\t &  0.9898       & \\cite{Tegel08a}\\\\\nBa122   & 0.9864\t &  0.9928       & \\cite{Rotter08a} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\noindent\n\nWe calculated total energies for different $b\/a$ ratios for different\nmagnetic models. For the nonmagnetic (NM), ferromagnetic (FM) (not\nshown) as well as the nearest-neighbour antiferromagnetic (NN-AFM)\n(Fig.~\\ref{dist}, upper panel) patterns, lowest total energy occurred\nat $a = b$, indicating that for these patterns the tetragonal\nstructure is more stable than the orthorhombic structure. Inclusion of\nspin-orbit effects did not change this result. The tetragonal to\northorhombic distortion is obtained only for the SDW pattern as\ndisplayed in the lower panel of Fig.~\\ref{dist}. Hence, LSDA\ncalculations clearly show that the SDW state is necessary for the\ntetragonal-to-orthorhombic transition to take place. The size of this\neffect, namely the deviation of the calculated $b\/a$ ratio from unity,\ndepends on the size of the cation, the ratio being smallest in\nBaFe$_2$As$_2$, and largest in CaFe$_2$As$_2$.  The values obtained\nvia LSDA are collected in Table~\\ref{bovera} and compared to the\nexperimental reports. The trend obtained in LSDA fits well with the\nexperimental $b\/a$ ratios with the exception of CaFe$_{2}$As$_{2}$.\nExperimentally, CaFe$_{2}$As$_{2}$ crystals are shown to have complex\nmicrostucture properties. Recent studies using transmission electron\nmicroscopy (TEM) \\cite{Ma2008} have shown a pseudo-periodic modulation\nand structural twinning arising from tetragonal to orthorhombic\ntransition only in CaFe$_{2}$As$_{2}$ but not in Sr or Ba $122$\nsystems. A structural twinning hinders the correct estimation of the\nlattice parameters and thereby may explain the experimental deviation\nin the trend of $b\/a$ ratio for CaFe$_{2}$As$_{2}$ with respect to\nLSDA.\n\nIn all four compounds the coupling along the shorter in-plane axis is\nFM in agreement with experimental findings. Although these results are\nrobust with respect to details of structure and calculations, the\npreferred direction of the spins are found to be quite sensitive. We\nperformed fully-relativistic calculations for $A =$ Ca, Sr and Ba\ncases using (i) SDW with $Q =$ [1 0 0], (ii) SDW with $Q =$ [1 0 1].\nThe latter SDW pattern requires the doubling of the $c$ lattice\nparameter and the corresponding calculations are highly time\nconsuming.  Therefore, only the cartesian axes are considered for\npossible spin orientations, and the structural data corresponding to\nthe minima in Fig.~\\ref{dist} are used.  In the Sr122 case for both\nSDW patterns we find the direction of the AFM coupling (along the\nlonger $a$-axis) as the easy axis in agreement with the neutron\nscattering study result \\cite{Jesche08a}.  However, in Ca and Ba 122\ncases different axes in the $(a,b)$ plane are found as easy axes for\ndifferent SDW patterns. The longer $a$-axis, is the easy axis for Ba\n(Ca) 122 for $Q =$ [1 0 0] ($Q =$ [1 0 1]); the shorter axis, $b$, the\ndirection of FM coupling, for Ba (Ca) 122 for $Q =$ [1 0 1] ($Q =$ [1\n  0 0]).  Since the involved energy differences are tiny (of the order\nof 15--30 $\\mu$eV per atom), a satisfactory resolution of this issue\nrequires further study.  Experimental study on\nBa122 \\cite{Huang08} has been able to determine that spins lie along\nthe longer $a$-axis.\n\n\\subsubsection{{\\bf Plasma frequency and effective dimensionality}}\n\nBand structure calculations can provide information on the ``effective\ndimensionality'' in a compound through various means, such as\ndispersionless (flat) energy bands along certain symmetry lines, van\nHove singularities in DOS, etc. A simple quantitative measure,\nhowever, can be obtained by computing plasma frequencies along the\nmain unit cell axes. For all of the $122$ compounds as well as the\nrepresentative compounds for the other families, plasma frequencies\nare calculated by nonmagnetic calculations using the experimental\nstructural data of the tetragonal phase.  In Table~\\ref{plasma} we\npresent the ratio of the in-plane plasma frequency\n$(\\omega_{\\mathrm{P}}^a = \\omega_{\\mathrm{P}}^b)$ to the plasma\nfrequency along the $c$-axis $(\\omega_{\\mathrm{P}}^c)$. One notices\nthat for the $122$ family the results are in line with expectations:\nthe compound is least anisotropic (more 3D-like) for the smallest\ncation (Ca), and strongly anisotropic (less 3D-like) for the largest\ncation (Ba). Additionally, the $1111$ and $1111^{\\prime}$ families are\nseen to be much more anisotropic (more 2D-like) than all of the\nothers.\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{\\label{plasma}Ratio of the in-plane plasma frequency\n  $\\omega_{\\mathrm{P}}^{a}$ to the out-of-plane plasma frequency\n  $\\omega_{\\mathrm{P}}^{c}$ for various members of the Fe-based\n  superconducting systems. The observed trend in the plasma frequency\n  ratios follows the trend in the ratios of the (inter-layer to\n  intra-layer) Fe-Fe distances,\n  $d_{c}^{\\mathrm{Fe-Fe}}$\/$d_{a}^{\\mathrm{Fe-Fe}}$. As we go down the\n  column, the systems go from being more 2D towards being more 3D. The\n  maximum superconducting transition temperature\n  $T_{c}^{\\mathrm{max}}$ obtained either via doping or pressure is\n  also collected in the last column. The trend in\n  $T_{c}^{\\mathrm{max}}$ also follows the trend in\n  $\\omega_{\\mathrm{P}}^{a}\/\\omega_{\\mathrm{P}}^{c}$, with decreasing\n  temperatures when the systems become more 3D. }\n\\begin{tabular}{ccccc}\n                  &$\\omega_{\\mathrm{P}}^{a}\/\\omega_{\\mathrm{P}}^{c}$& $c\/a$ &  $d_{c}^{\\mathrm{Fe-Fe}}$\/$d_{a}^{\\mathrm{Fe-Fe}}$ & $T_{c}^{\\mathrm{max}}$ (K)\\\\\n\\hline\nSrFeAsF           &     19.892                &2.2426 &  3.1715  & 56 \\cite{Wu08b} \\\\\nLaOFeAs           & \t8.9467\t\t      &2.1656 &\t 3.0626  & 55 \\cite{Ren08a} \\\\\nFeSe              &\t4.1119                &1.4656 &  2.0727  & 27 \\cite{Mizuguchi08a} \\\\\nLiFeAs            &   \t3.2181                &1.6785 &\t 2.3738\t & 18 \\cite{WangXC08a} \\\\\nBaFe$_{2}$As$_{2}$&   \t3.2926\t              &3.2850 &  2.3228  & 38 \\cite{Rotter08b} \\\\\nSrFe$_{2}$As$_{2}$&   \t2.8329\t              &3.1507 &  2.2279  & 38 \\cite{GFChen08a,Sasmal08a} \\\\\nCaFe$_{2}$As$_{2}$&\t1.3953                &3.0287 &  2.1416  & 20 \\cite{Wu08a} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn the case of iron arsenides, in a rather simplified picture, one\nexpects the plasma frequency ratio to be proportional to the ratio of\nthe shortest interlayer $d_c^{\\mathrm{Fe-Fe}})$ to the shortest\nintralayer Fe-Fe distances $(d_a^{\\mathrm{Fe-Fe}})$. In terms of\nlattice parameters the distance ratio\n$d_c^{\\mathrm{Fe-Fe}}\/d_a^{\\mathrm{Fe-Fe}}$ is $c\/\\sqrt2 a$ for\n$122$'s and $\\sqrt2 c\/a$ for the others. Table~\\ref{plasma} shows that\nthese two ratios are largely correlated with apparently the exception\nof FeSe \\cite{footnote3}.\n\n\\subsubsection{{\\bf Effect of the As $z$ position and magnetic moments from\n LDA:}}\n\\label{Aszpos}\n\nIn any first-principles study of a magnetic system, an essential\naspect is the comparison of the computed magnetic moments with the\nexperimentally deduced ones.  This standard procedure proves to be\nquite tricky in these FeAs-based compounds due to the unexpected\nsensitivity of the magnetism to the As $z$ position \\cite{Mazin08a}.\nWe performed a series of calculations \\cite{footnote1} for Ca, Sr and\nBa 122 systems for FM, NN-AFM and SDW spin patterns using the\nexperimental volume (ambient pressure, below $T_{N}$) both with $a =\nb$ and $a \\neq b$ and optimizing the Fe-As distance (via As $z$\nposition) for each case.  Table~\\ref{mom} compares the Fe moments\ncomputed for the SDW pattern (the lowest-total-energy spin pattern\namong those considered) with the experimental values obtained using\nneutron diffraction and muon spin rotation ($\\mu$SR).  In comparison\nto the situation in the 1111 systems, here for the 122 systems, the\nagreement between theory and experiment is seen to be better.\nHowever, the computed Fe magnetic moment is found to increase from Ca\nto Ba 122, whereas the trend is just the opposite according to the\nestimation of moments from $\\mu$SR results. The moments obtained from\nneutron diffraction experiments are more reliable and remain rather\nconstant for the three $122$ systems considered here.  The influence\nof the orthorhombic distortion on the calculated moments is quite\nnegligible.\n\nThe better agreement between theory and experiment regarding the Fe\nmagnetic moment in the case of 122 systems can be understood as\nfollows.  The computed \\textit{vs} measured magnetic moment\ndiscrepancy in the 1111 systems is usually explained to be a result of\nlarge spin fluctuations \\cite{Mazin08b}.  It is also known that spin\nfluctuation effects are reduced when going from 2D systems to 3D\nsystems.  Hence, the results presented in Table~\\ref{mom} provide\nadditional support for the effective dimensionality considerations\ndescribed above: the 122 systems have a more pronounced 3D nature than\nthe 1111 systems.  Since the description of spin fluctuation effects\nis insufficient in LSDA, the computed values are immune to such\neffects, while, of course, the values deduced from experiments do\nreflect these effects.  Furthermore, since the Ba 122 system is more\n2D-like than the Ca 122 system (\\textit{cf.}  Table~\\ref{plasma}), it\nis expected to exhibit a smaller Fe moment (stronger spin fluctuation\neffects), and this is in agreement with the $\\mu$SR results.\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption{\\label{mom}Comparison of the magnetic moments in\n  $\\mu_{\\mathrm{B}}$ per atom calculated using LSDA with the\n  experimental values obtained via $\\mu$SR (a local probe) and neutron\n  diffraction measurements. In LSDA, the moments were calculated for\n  the SDW pattern for the Fe spins using a tetragonal lattice ( $a =\n  b$ ) and orthorhombic lattice ( $a \\neq b$ ). The Fe-As distance has\n  been optimized for all the calculations reported in this table.}\n\\begin{tabular}{ccccc}\n            &  \\multicolumn{2}{c}{LDA - SDW} &   $\\mu$SR    & Neutron  \\\\\n\\cline{2-3}\n            &   $a = b$  &  $a \\neq b$ &                  & diffraction \\\\\n\\hline\nCa122       & 0.818    &   0.875     & 0.9 \\cite{TGoko08a}   & 0.80 \\cite{Goldman08a} \\\\\nSr122       & 1.1      &   1.13      & 0.8 \\cite{TGoko08a}   & 0.94 \\cite{Zhao08a} \\\\\nBa122       & 1.12     &   1.17      & 0.5 \\cite{TGoko08a}   & 0.87 \\cite{Matan08a} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe interplay between the Fe-As distance and different magnetic\nlong-range orders is illustrated in Fig.~\\ref{Asz} for\nSrFe$_{2}$As$_{2}$.  Similar results are found for the other $122$\nsystems.  The experimental value of the Fe-As distance in\nSrFe$_{2}$As$_{2}$ in the orthorhombic phase is reported to be 2.391\n\\AA\\ at 90 K \\cite{Tegel08a}.  Using the FM or the NN-AFM spin pattern\nproduces a minimum in energy at around 2.31 \\AA\\ along with a complete\nloss of the Fe magnetic moment.  In the SDW pattern, the optimum value\nof the Fe-As distance is 2.327 \\AA, only slightly larger than that\nobtained using the FM or NN-AFM pattern, but the Fe magnetic moment is\nstill 1.13 $\\mu_{\\mathrm{B}}$.\n\n\\begin{figure}[htb]\n\\begin{center}\\includegraphics[%\n  clip,\n  width=8.5cm,\n  angle=0]{NN_SDW_As_z_comp.eps}\\end{center}\n\\caption{\\label{Asz} Energy and Fe magnetic moment as a function of\n  the Fe-As distance using different spin patterns for\n  SrFe$_{2}$As$_{2}$ at the experimental volume around 90 K\n  \\cite{Tegel08a}. Optimization using FM and NN-AFM pattern leads to a\n  nonmagnetic solution, while the SDW pattern stabilizes with a Fe\n  moment of 1.13 $\\mu_{B}$. The energy curves have been shifted by\n  setting the minimum energy value to zero. The dashed vertical line\n  refers to the experimental Fe-As distance obtained from\n  Ref. \\cite{Tegel08a}. The arrows indicate the position of the energy\n  minima.  }\n\\end{figure}\n\nThe As-$z$ position is most likely one of the key issues for\nunderstanding of the iron pnictides.  Present day density functional\ntheory based calculations using LDA (described above) and also GGA\n\\cite{Mazin08a,Yildirim08a} are not able to reproduce all the\nexperimental findings consistently. Since the magnetism and therefore\nthe superconductivity crucially depends on this structural feature and\nthe related accurate description of the Fe-As interaction, the\nimprovement of the calculations in this respect may offer the key to\nthe understanding of superconductivity in the whole family.\n\n\n\\subsection{{\\bf Effects of pressure}}\n\\label{pressure}\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\epsfig{file=EvsVcomp.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\epsfig{file=EvsVwithca.eps, clip=, width=\\linewidth}\n    \\end{minipage}\n\\caption{\\label{EvsV}{\\bf Left:} Energy as a function of volume for\n  $A$Fe$_{2}$As$_{2}$ systems, with $c\/a$ ratio optimized. The two\n  curves correspond to the antiferromagnetically ordered Fe spins in\n  the SDW pattern with a non-zero moment (red circles) and zero moment\n  (blue triangles). The kink at the intersection of these two curves\n  is caused by a collapse of the $c\/a$ ratio upon pressure, which\n  happens at the juncture when the systems lose their Fe moments and\n  become non-magnetic. The $c\/a$ ratio collapse is most pronounced in\n  CaFe$_{2}$As$_{2}$ due to the small size of the Ca ion and as the\n  size of the $A$ ions increases, this feature becomes more and more\n  subtle. {\\bf Right:} The evolution of the $c\/a$ ratio collapse for\n  CaFe$_{2}$As$_{2}$. Notice the emergence of the double minima.  }\n  \\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\begin{minipage}[t]{0.46\\linewidth}\n      \\raisebox{1cm}{\\includegraphics[clip=, width=\\linewidth, angle=-90]{Volvsmom_sdw.eps}}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.52\\linewidth}\n      \\caption{\\label{momvsV} Volume dependence of the Fe moments\n        corresponding to the energy-volume curves in\n        Fig.~\\ref{EvsV}. The quenching of the spin magnetic moment is\n        highly abrupt for the compounds with pronounced $c\/a$ ratio\n        collapse. The Fe moment in BaFe$_{2}$As$_{2}$ goes to zero\n        smoothly reflecting the much smoother $c\/a$ variation obtained\n        for this compound. }\n    \\end{minipage}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}[b]\n  \\begin{center}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\raisebox{-7cm}{\\includegraphics[clip=, width=1.3\\linewidth, angle=0]{dos_pressure.eps}}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.5\\linewidth}\n      \\caption{\\label{dospressure} Total DOS per cell as a function of\n        reduced volume for SrFe$_{2}$As$_{2}$ corresponding to the\n        energy-volume curves in Fig.~\\ref{EvsV}. In the SDW pattern,\n        the total DOS for both the spin-up and spin-down channels are\n        the same. Therefore, we show here the DOS from only one of the\n        spin channels. The DOS at the Fermi level $E_{F}$ at first\n        decreases with decreasing volume, but later starts to increase\n        after the Fe spin magnetic moment is quenched. }\n    \\end{minipage}\n  \\end{center}\n\\end{figure}\n\n\n\nAs mentioned above, superconductivity in the FeAs based systems can be\nachieved via doping of charge carriers.  This kind of chemical\nsubstitution is quite convenient, but changes the electronic structure\nof the doped systems in a non-trivial way as compared to the undoped\nsystems. Using external pressure as a probe on these systems creates a\nsimilar effect as doping, without the added complexity.  Application\nof hydrostatic pressure suppresses both the tetragonal to orthorhombic\ndistortion and the formation of the SDW, and leads to the onset of\nsuperconductivity in a similar fashion as charge doping.  All the\nparent members of the $122$ family have been reported to superconduct\nor show signs of its onset under pressure\n\\cite{Torikachvili08a,Kreyssig08a,Kumar08a,Miclea08a,Alireza08a}.\nCa122 was originally reported to superconduct ($T_{c}$ $\\approx$ 10 K)\nat 0.4 GPa pressure \\cite{Torikachvili08a,Kreyssig08a} but recent work\nby Yu and collaborators \\cite{Yu08a} do not observe any signature of\nbulk superconductivity and suggest a possible phase separation due to\nnon-hydrostatic conditions. Similarly, Sr122 was first reported to\nsuperconduct at 2.8 GPa pressure with a $T_{c}$ of 27 K\n\\cite{Alireza08a}, while on the contrary Kumar {\\it et al.}\n\\cite{Kumar08a} did not observe a zero-resistance state up to 3 GPa\npressure. A sharp drop in resistivity above 2 GPa pressure was\nreported for Eu122, suggesting the onset of superconductivity and,\nmore interestingly, signatures of possible re-entrant\nsuperconductivity.  Though there exist conflicting reports of the\ntransition pressure and possible phase separation in the sample under\npressure, it is worthwhile to explore the evolution of the electronic\nstructure and magnetism as a function of pressure.  Pressure studies\n\\cite{Kreyssig08a} on the CaFe$_{2}$As$_{2}$ system report a\nsignificant $c\/a$ collapse along with a structural transition\n(orthorhombic to tetragonal) under modest pressures of less than 0.4\nGPa, while no such collapse has been reported for the other compounds.\nX-ray diffraction refinements carried out at 180 K for\nSrFe$_{2}$As$_{2}$ \\cite{Kumar08a} observe an orthorhombic to\ntetragonal transition above 3.8 GPa, but no collapse of the $c\/a$\nratio is observed.  Band structure calculations allow for the study of\nsuch features up to very large pressures, that might not be easily\nattainable through experiments.  We have calculated energy as a\nfunction of volume for all the four systems in the $122$\nfamily. Firstly, we wanted to investigate the possibility of a $c\/a$\ncollapse for each member of the $122$ family.  Therefore, we\ncalculated energy as a function of volume using the SDW pattern and\noptimizing only the $c\/a$ ratio at each volume. The internal As-$z$\nparameter was kept fixed at the experimental (room temperature)\nvalue. The results from these calculations are collected in\nFig.~\\ref{EvsV}.  Surprisingly, all the four $122$ systems have a kink\nin the energy-volume curve caused by a non-continous change in the\n$c\/a$ ratio, which happens at the juncture when the systems lose\ntheir Fe moments and become non-magnetic (see Fig.~\\ref{momvsV}).  The\nsharpness of the kink is largest for Ca122 and decreases with the\nincreasing size of the $A$ ion. Such an $A$-ion size effect was also\nobserved and discussed in section 3.2.1 when investigating the size of\nthe orthorhombic $b\/a$ ratio.  Our result obtained from LSDA (via a\ncommon tangent construction) for CaFe$_{2}$As$_{2}$ is in excellent\nagreement with the previously reported \\cite{Kreyssig08a} experimental\ndata, volume collapse: $\\delta V^{\\mathrm{LDA}}$ $\\approx$ 4.7\\%,\n$\\delta V^{\\mathrm{exp}}$ $\\approx$ 5\\%; ratio collapse:\n$\\delta$($c\/a$)$^{\\mathrm{LDA}}$ $\\approx$ 9.8\\%,\n$\\delta$($c\/a$)$^{\\mathrm{exp}}$ $\\approx$ 9.5\\%. Experimentally\n\\cite{Kreyssig08a} a pressure of $\\sim$ 0.3 GPa induces a transition\nfrom the orthorhombic phase to a collapsed non-magnetic tetragonal\nphase for CaFe$_{2}$As$_{2}$.  Scaling the volumes of the different\n$122$ systems with respect to the experimental values\n($V\/V_{\\mathrm{exp}}$), one observes that the kink in the\nenergy-volume curve for the other three $122$ systems happens at lower\nvolume ratios (or larger pressures) as compared to the Ca122. It\nshould be worthwhile to investigate this structural feature\nexperimentally by applying higher pressures to the Sr, Ba and Eu $122$\nsystems.\n\nChanges in the electronic structure as a function of reduced volumes\nwere carefully monitored. Shown in Fig.~\\ref{dospressure} are the\ntotal DOS for SrFe$_{2}$As$_{2}$ at selected volumes. Similar results\nare obtained for other $122$ systems. The DOS at $E_{F}$ decreases\ngradually at first for up to 10\\% volume reduction with respect to the\nexperimental volume. The Fe ions continue to carry a magnetic moment\nthough the actual values are quite reduced. Upon further reduction of\nthe volume, the spin moments get quenched and the DOS at $E_{F}$ begin\nto increase and the system becomes non-magnetic. At the experimental\nvolume, the net moments on the various Fe orbitals are quite similar;\nwith the Fe-3$d_{x^{2}-y^{2}}$ having a slightly larger value than the\nother orbitals. With the application of pressure, the net moments of\nall the five $d$ orbitals decrease in a similar fashion and tend to\nzero.\n\n\\begin{figure}[t]\n\\begin{center}\\includegraphics[%\n  clip,\n  width=10cm,\n  angle=0]{EvsV_fullopt.eps}\\end{center}\n\\caption{\\label{fullopt} {\\bf Top panel:} Energy as a function of\n  volume after a full relaxation of all the parameters for\n  SrFe$_{2}$As$_{2}$ and CaFe$_{2}$As$_{2}$. Contrary to the results\n  depicted in Fig.~\\ref{EvsV}, we now no longer observe any kink at\n  the juncture when the Fe ions lose the magnetic moment\n  (corresponding data point is indicated using an arrow). {\\bf Bottom\n    panel:} Fe magnetic moment as a function of volume for the Ca and\n  Sr 122 systems. The results shown here are different from the moment\n  values collected in Table.~\\ref{mom}, because no $c\/a$ optimization\n  was carried out for the latter.  }\n\\end{figure} \n\nAnother important feature that needs to be addressed in regard to the\nenergy-volume curves is the serious underestimation of the equilibrium\nvolume within LDA.  Generally, equilibrium volumes obtained from LDA\nare smaller within up to 8\\% of the experimentally reported values. In\nthe case of 122 systems using SDW pattern, we obtain values that are\n13\\%, 15\\%, 10\\%, and 13\\% smaller than the experimental reports for\nCa, Sr, Ba and Eu 122 systems respectively. Moreover, at the LSDA\nequilibrium volume, contrary to the experimental reports, all the\nparent compounds are computed to be nonmagnetic.  The reason for this\ndiscrepancy is unclear.  In the previous sections we discussed the\npronounced sensitivity of the Fe moments to the various structural\nparameters in these FeAs systems.  Although with partial optimization\n(fixed As-$z$ parameter) we have satisfactorily accounted for the\nexperimentally observed $c\/a$ collapse in CaFe$_{2}$As$_{2}$, it is\nnecessary to find out what the ultimate LSDA solution is regarding the\ngeometrical structure and magnetism in the 122 systems. Consequently,\nfor Ca and Sr 122 systems and using the SDW pattern, at each volume we\nhave optimized all three free structural parameters in the following\norder: 1) As-$z$ position, 2) $c\/a$ ratio, 3) $b\/a$ ratio.  This\nsequence of steps has been repeated until the energies obtained are\nconverged to an accuracy of 10$^{-6}$ eV.  Collected in\nFig.~\\ref{fullopt} are the energy-volume curves for the Ca and Sr\n$122$ systems.  The equilibrium volume obtained after a full\noptimization is only slightly larger than the values obtained after\njust a $c\/a$ optimization (see Fig.~\\ref{EvsV}).  Contrary to the\nresults depicted in Fig.~\\ref{EvsV}, we now no longer observe any kink\nat the juncture when the Fe ions lose their magnetic moments. The\nloss of moment for Sr122 is more gradual than for Ca122.  As volume is\ndecreased (higher pressures are applied) the Fe spin magnetic moments\ntend to zero while the orthorhombic distortion ratio $b\/a$ tends to\nunity so that at increased pressures the tetragonal lattice is\nfavored. This observation of the lattice structure preferring the\ntetragonal symmetry when the Fe ions become nonmagnetic is consistent\nwith our previous results in Sec.~\\ref{str}, and reaffirms the\nintimate connection between structure and magnetism for the 122\nsystems.  Optimizing the As $z$ parameter again tends to confirm\ncertain experimental findings (the connection between SDW magnetic\npattern and orthorhombic distortion) but not all (for example, lack of\n$c\/a$ collapse under pressure for CaFe$_{2}$As$_{2}$).  This again\nre-affirms the need for a correct description of the Fe-As interplay\nto obtain consistent results.\n\n\\subsection{{\\bf Effects of charge doping} }\n\n\\begin{figure}[htb]\n\\begin{center}\\includegraphics[%\n  clip,\n  width=8.5cm,\n  angle=0]{picsinfile.ps}\\end{center}\n\\caption{\\label{vca} Results from the VCA calculations. Magnitude of\n  the Fe moment as a function of charge doping (both on Sr site as\n  well as Fe site) for different ordering patterns in\n  SrFe$_{2}$As$_{2}$.  When {\\bf $x$} is positive: electron doping;\n  when {\\bf $x$} is negative: hole doping.  The filled symbols and the\n  open symbols indicate doping of the $A$ site and the Fe site\n  respectively. `D' represents the dopant. Very different effects are\n  observed when doping charge carriers on the Sr or Fe site. Magnetism\n  is weakened when electrons are substituted on the Fe site, while\n  strengthened when electrons are substituted on the Sr site. The\n  relative trend between the different ordering patterns remains the\n  same.  We have used the experimental lattice parameters (at\n  $\\approx$ 300 K) and As-$z$ value for all the calculations. }\n\\end{figure}\n\n\\begin{figure}[htb]\n  \\begin{center}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\epsfig{file=co_vs_la_dos.ps, clip=, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\epsfig{file=la_series_dos.ps, clip=, width=\\linewidth}\n    \\end{minipage}\n\\caption{\\label{covca}{\\bf Top Left:} Non magnetic total and Fe\n  orbital resolved DOS from a VCA calculation for a 15\\% electron\n  doping on the Fe site in Sr122. Upon electron doping, the DOS\n  remains unchanged and displays a rigid-band-like behaviour.  {\\bf\n    Bottom Left:} Non-magnetic total and Fe orbital resolved DOS for a\n  30\\% electron doping on the Sr site in Sr122. Addition of electrons\n  changes the shape of the DOS close to $E_{F}$ drastically. A\n  pronounced peak starts to appear close to $E_{F}$ which tends to\n  destabilize the system. The changes in the total DOS for various\n  doping concentrations are shown in the right panel.  {\\bf Right:}\n  Non magnetic total DOS as a function of additional electrons on the\n  Sr site in Sr122.  }\n  \\end{center}\n\\end{figure}\n\nIn order to understand the influence of charge doping (both electrons\nand holes) on the electronic structure and henceforth the magnetism,\nwe performed total energy calculations using the VCA for three\ndifferent spin patterns: FM, NN-AFM and SDW.  We considered both\nelectron and hole doping on the $A$ site as well as the Fe site.  Our\nresults for the changes in the Fe moment in SrFe$_{2}$As$_{2}$ are\ncollected in Fig.~\\ref{vca}. Similar results were obtained for other\nmembers of the 122 family. We observe very different effects depending\non the sign (electrons or holes) and site (Sr or Fe) of the doping.\nRegardless of the choice of magnetic ordering, electron doping on the\nFe (Sr) site weakens (enhances) the magnetism.  This behaviour can be\nexplained by analyzing how the nonmagnetic electronic structure\nchanges with electron doping (see Fig.~\\ref{covca}).  Electron doping\non the Fe site (left upper panel of Fig.~\\ref{covca}) results in DOS\nvery similar to that of the undoped case, main effect being the\n$E_{F}$ moved toward higher energies to accommodate the added\nelectrons.  On the other hand, electron doping on the Sr site changes\nthe resultant DOS drastically (see left lower and right panel of\nFig.~\\ref{covca}). Most of the major changes to DOS occur in the close\nvicinity of $E_{F}$ giving rise to pronounced peaks at the $E_{F}$.  A\nlarge value of DOS at the Fermi level, $N(E_F)$, is usually a sign of\ninstability for an electronic system.  The system can lower $N(E_F)$\nby, for example, developing a long-range magnetic order provided the\nStoner criterion is satisfied.  The larger values of the computed Fe\nmagnetic moments may reflect such an increased instability to magnetic\norder.  Additionally, with reservations for possible thermodynamical\nconsiderations, appearance of this feature may explain why La doped\n122 samples could not be synthesized until now.\n\nSubstitution of holes on the Sr site does not introduce significant\nchanges to the Fe magnetic moment.  Substitution of holes on the Fe\nsite tends to enhance magnetism for both AFM patterns whereas for the\nFM pattern the magnetism vanishes beyond a critical level of\ndoping. However, this feature for the FM spin pattern is of no\nsignificance, because energetically it lies above both of the AFM\npatterns at all levels of doping.\n\n\\subsection{{\\bf Electric field gradient}}\n\nNuclear magnetic resonancce (NMR) is a local probe that is extremely\nsensitive to certain details of the structure. Since the As $z$\nposition is a key determinant of many of the electronic properties of\nthe FeAs systems, the quadrupole frequency $\\nu_{Q}$ from NMR\nmeasurements can provide a direct measure to the Fe-As interaction.\nTheoretically, $\\nu_{Q}$ can be obtained by calculating the electric\nfield gradient (EFG).  The EFG is defined as the second partial\nderivative of the electronic potential $v(\\vec r)$ at the position of\nthe nucleus\n\\begin{eqnarray}\nV_{ij}&=&\\left(\\partial_i\\partial_j v(0) - \\frac{1}{3}\\Delta \\delta_{ij} \\right)\n\\Delta v(0).\n\\end{eqnarray}\nThis traceless and symmetric tensor of rank 2 is described in the\nprincipal axis system by the main component $V_{zz}$ and the asymmetry\nparameter $\\eta = (V_{xx}-V_{yy})\/V_{zz}$.  $V_{zz}$ is per definition\nthe component with the largest magnitude $|V_{zz}|\\geq|V_{yy}| \\geq\n|V_{xx}|$ and is not necessarily parallel to the $z$-axis of the\ncrystal.  From these two parameters ($V_{zz}$ and $\\eta$) and the\nquadrupole moment for $^{75}$As $Q=(0.314\\pm0.006)$~b \\cite{pyykko}\nthe quadrupole frequency $\\nu_Q$ for $^{75}$As (with a nuclear spin of\n$I=3\/2$) can be calculated \\cite{abragam}\n\\begin{eqnarray}\n\\label{nuQ}\n\\nu_Q=\\frac{eQV_{zz}}{2h}\\sqrt{1+\\frac{\\eta^2}{3}}.\n\\end{eqnarray}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[clip,width=8cm,angle=0]{As_z3.eps}\n\\end{center}\n\\caption{\\label{EFG1}Dependence of the EFG for As on the As $z$\n  position. $\\Delta z = z - z_{\\mathrm{exp}}$. Different symbols show\n  different calculations: cross = nonmagnetic with tetragonal\n  symmetry, full circle = nonmagnetic with orthorhombic symmetry\n  (almost identical to tetragonal symmetry for SrFe$_2$As$_2$ and\n  therefore not shown) and empty circle = magnetic (SDW pattern) with\n  orthorhombic symmetry. The total energy minimum is marked by an\n  arrow for each nonmagnetic curve. The error bars show the\n  experimental results for the tetragonal phase (at 250~K for\n  CaFe$_2$As$_2$ and at 200~K for BaFe$_2$As$_2$ (the two error bars\n  indicate the experimentally unknown sign of the EFG)).}\n\\end{figure}\n\nThe experimental lattice parameters, including the As $z$ position for\nthe calculation of the EFG for As in CaFe$_2$As$_2$, SrFe$_2$As$_2$\nand BaFe$_2$As$_2$ were obtained from Ref.  \\cite{Kreyssig08a},\n\\cite{Sr122GiPa}, \\cite{Rotter08a} respectively. For the parent\ncompounds we investigated the influence of the As $z$ position, the\nstructural phase transition, the magnetism and the pressure on the\nEFG. We also investigated the effects of doping on the EFG.\n\nFirst we focus on the As $z$ dependence of the EFG.  Just as the\ncomputed Fe magnetic moment, whose value show a strong dependence on\nthe As $z$ position (see Sec.~\\ref{Aszpos}), the EFG is also found to\ndisplay a strong As $z$ dependence.  The EFG increases strongly for\nall three compounds, as the Fe-As distance decreases, see\nFig.~\\ref{EFG1}. In case of CaFe$_2$As$_2$, there is a minimum in the\nEFG for a displacement of roughly $\\Delta z=-0.1$~{\\AA} from the\nexperimental position, while for larger Fe-As distances the EFG\nincreases again. The same trend is observed for the other two\ncompounds, middle and lower panels in Fig.~\\ref{EFG1}. They exhibit\nthe minimum in the EFG at about $\\Delta z=+0.05$~{\\AA}.  For all the\nthree parent compounds, the Fe-As distance at which the minimum in\ntotal energy occurs is smaller than that corresponds to the EFG\nminimum (see arrows in Fig.~\\ref{EFG1}).  For CaFe$_{2}$As$_{2}$ we\nobserve a good agreement between the calculated EFG at the\nexperimental As $z$ position (for 250~K) and the measured EFG at 250~K\n\\cite{Baek} (the light green symbol in the upper panel of\nFig.~\\ref{EFG1}).  In case of BaFe$_2$As$_2$ the magnitude of the\nmeasured EFG at 200~K is roughly $0.7\\times 10^{21}$~V\/m$^2$\n\\cite{Kitagawa}, while the sign is unknown since it cannot be\nextracted from NQR measurements (In Fig.~\\ref{EFG1} the experimental\nEFG values with both signs are shown).  The calculated $V_{zz}$ for\nthe experimental As $z$ position is $-1.1\\times 10^{21}$~V\/m$^2$. If\nthe experimental EFG is negative, reasonable agreement between\nexperiment and calculation is obtained.  In a preliminary measurement\nfor SrFe$_{2}$As$_{2}$ the quadrupole frequency $\\nu_Q$ was determined\nto be positive and less than 2~MHz \\cite{peter} and this is also\nconsistent with the calculated EFG of 0.8~MHz at the experimental As\n$z$ position.  For members of the $1111$ family; LaFeAsO and NdFeAsO:\nthe calculated EFG for the optimized As $z$ position agreed better\nwith the experimental EFG \\cite{Lapaper,Ndpaper}.  Our results for\nthree representative members of the $122$ family as shown above follow\na different trend: the calculated EFG using the experimental As $z$\nposition agree better with the measured EFG values.\n\nTo study the influence of the orthorhombic distortion, but without the\ninfluence of magnetism we perform non-magnetic calculations both in\ntetragonal and orthorhombic symmetry. The orthorhombic splitting of\nthe axes in the $(a,b)$ plane has a rather small influence on the\nEFG. The EFG is larger for the orthorhombic symmetry for small Fe-As\ndistances, i.e. $\\Delta z<-0.1$~{\\AA}.  In case of SrFe$_2$As$_2$, the\neffect of the orthorhombic splitting is so small, that the\northorhombic EFG curve in Fig.~\\ref{EFG1} is not shown (see middle\npanel of Fig.~\\ref{EFG1}). In case of BaFe$_2$As$_2$, we observe\nsimilar behaviour as for CaFe$_2$As$_2$. The tetragonal and\northorhombic EFG curves cross close to the EFG minimum and the EFG is\nlarger for the orthorhombic symmetry for smaller Fe-As distances,\ni.e. $\\Delta z<+0.1$~{\\AA}.  For all three compounds we find that\n$V_{zz}$ is parallel to the crystallographic $z$-axis for the\nnon-magnetic calculations in both the tetragonal and orthorhombic\nsymmetry.\n\n\n\\begin{figure}[b]\n  \\begin{center}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\epsfig{file=MagnetismSr.ps, clip=, angle=-90, width=\\linewidth}\n    \\end{minipage}\n    \\begin{minipage}[t]{0.48\\linewidth}\n      \\epsfig{file=MagnetismBa.ps, clip=, angle=-90, width=\\linewidth}\n    \\end{minipage}\n\\caption{\\label{EFG2}{\\bf Left:} The three components of the EFG\n  tensor for SrFe$_2$As$_2$ in the orthorhombic SDW phase as a\n  function of the As $z$ position. $\\Delta z = z -\n  z_{\\mathrm{exp}}$. The component of the EFG, parallel to the\n  crystallographic $x$-axis is shown by triangles up, the component\n  parallel to the crystallographic $y$-axis by squares and the one\n  parallel to the crystallographic $z$-axis by solid\n  circles. $V_{zz}$, the largest one of these three, is marked by a\n  large shaded circle for each As $z$ position.  {\\bf Right:} The\n  three components of the EFG tensor for BaFe$_2$As$_2$ in the\n  orthorhombic SDW phase as a function of the As $z$ position. The\n  rest of the notation is the same as the left panel. }\n  \\end{center}\n\\end{figure}\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{$V_{zz}$ in $10^{21}$~V\/m$^2$ for the nonmagnetic and\n  different magnetic orders, all in orthorhombic phase.}\\label{EFGtable}\n\\begin{tabular}{lcccc}\n compound &  NM &  FM & NN--AFM  & SDW\n\\\\\n\\hline\nCaFe$_2$As$_2$ & 2.6 & 2.4 & 2.7 & 3.1\n\\\\\nSrFe$_2$As$_2$ & 0.2 & 0.3 & 0.2 &-1.3\n\\\\\nBaFe$_2$As$_2$ & -1.1 & -1.0 & -1.3 &+1.3\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nInvestigation of the influence of the magnetism on the EFG in the\northorhombic symmetry show that, FM or NN--AFM ordering of the Fe\natoms does not change the EFG much, however, the SDW order has a huge\ninfluence on the EFG, \\textit{cf.} Table~\\ref{EFGtable}. For all the\nthree systems, as the Fe-As distance is decreased, the magnetic moment\nis reduced and finally tends to zero. At this displacement value, the\nSDW EFG curves smoothly join the non-magnetic orthorhombic EFG curves\nas one would expect.  In Fig.~\\ref{EFG1} the component of the EFG,\nthat is parallel to the $z$-axis of the crystal is shown. As mentioned\nbefore, $V_{zz}$ is found to be parallel to the crystallographic\n$z$-axis in all non-magnetic calculations. For the magnetic SDW phase\nthe same behaviour is observed for CaFe$_2$As$_2$, but not for\nSrFe$_2$As$_2$ and BaFe$_2$As$_2$.  For the latter two compounds,\n$V_{zz}$ changes the axis, \\textit{i.e.}, the axis along which EFG is\nthe largest changes as Fe-As distance is varied. For\nBaFe$_{2}$As$_{2}$, such a behaviour was also observed experimentally\n\\cite{Kitagawa} when going from the high-temperature nonmagnetic\ntetragonal phase to the low temperature SDW phase.  Unfortunately for\nCaFe$_2$As$_2$, only the quadrupole frequency parallel, and not\nperpendicular to the crystallographic $z$-axis is provided in\nRef.~\\cite{Baek}. The three diagonal components of the EFG tensor\n$V_{ii}$, which are parallel to the $x$, $y$ and $z$-axis of the\ncrystal, vary continuously as a function of the As $z$ position, as\ncan be seen in Fig.~\\ref{EFG2}. In case of SrFe$_2$As$_2$, $V_{zz}$ is\nparallel to the $x$-axis for a displacement of As between\n$+0.05$~{\\AA} and $-0.075$~{\\AA} (which includes the experimental As\n$z$ position) and parallel to the $z$-axis for a displacement between\n$-0.1$~{\\AA} and $-0.15$~{\\AA}.\nFor BaFe$_2$As$_2$ according to its definition as the largest\ncomponent $V_{zz}$ fluctuates between all the three different axes\n(Fig.~\\ref{EFG2}) In particular, at the experimental As $z$ position\n$V_{zz}$ is parallel to the $y$-axis.  We also observe that the\ncomponent parallel to the $x$-axis is very similar for both Sr and Ba\n122 compounds.  The components parallel to the $y$- and $z$-axis show\nthe same variation with As $z$ position, only the values for the two\ncompounds are shifted by an almost constant amount.\nFig.~3 in Ref.~\\cite{Baek} shows the temperature dependence of the\nquadrupole frequency $\\nu_Q$ for CaFe$_2$As$_2$: $\\nu_Q$ increases\ndrastically from 300~K to 170~K. At 170~K there is a large jump in the\nfrequency due to the orthorhombic SDW phase transition. Between 170~K\nand 20~K $\\nu_Q$ is rather constant.  The calculated EFGs correspond\nto lattice parameters at 250~K and 50~K. For these two temperatures\nthe quadrupole frequency (parallel to the crystallographic $z$-axis)\nis almost identical. This is in agreement with our result for the\nexperimental As $z$ position as seen in the upper panel of\nFig.~\\ref{EFG1}.\n\nWe also investigated the influence of pressure on the EFG.  $V_{zz}$\nfor different pressures was calculated for CaFe$_2$As$_2$\n\\cite{Kitagawa} and SrFe$_2$As$_2$ \\cite{Kumar08a} using the\nexperimental structural parameters reported as a function of pressure.\nOur result is shown in the inset of Fig.~\\ref{EFG3}. In case of\nCaFe$_2$As$_2$, the EFG increases when the applied pressure is\nincreased from 0~GPa to 0.24~GPa. For these pressures the structure is\nin the (orthorhombic) SDW phase. The next experimental pressure point\nis larger than the critical pressure of 0.3~GPa (Sec.~\\ref{pressure}),\nwhere the $c\/a$ collapse takes place. The structure changes into the\nnonmagnetic tetragonal phase and the calculated EFG increases\ndrastically from roughly 3 to 10$\\times 10^{21}$~V\/m$^2$.\nExperimentally, the applied pressure for SrFe$_2$As$_2$ was much\nhigher (up to 4~GPa) than for CaFe$_2$As$_2$, but no indications of a\ncollapsed phase was found till now. Contrary to the jump in the\ncalculated EFG at 0.3 GPa for CaFe$_{2}$As$_{2}$, EFG for\nSrFe$_{2}$As$_{2}$ increases monotonously without any kinks upto 4\nGPa.  It is worthwhile to measure the EFG for these systems to get a\nmore clear picture.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[clip,width=8cm,angle=-90]{Sr+BaDopingVCA+pressure.ps}\n\\end{center}\n\\caption{\\label{EFG3} EFGs calculated for doped CaFe$_2$As$_2$ (green\n  circles), SrFe$_2$As$_2$ (red squares) and BaFe$_2$As$_2$ (blue\n  diamonds) using VCA. Results obtained from a four-fold super cell\n  for {\\bf $x$} = -0.25 for SrFe$_2$As$_2$ (shaded orange squares) and\n  BaFe$_2$As$_2$ (shaded blue diamonds) in the nonmagnetic tetragonal\n  phase are also shown. {\\bf Inset:} Dependence of the EFG on pressure\n  for CaFe$_2$As$_2$ (left) and SrFe$_2$As$_2$ (right). The latter is\n  in the (nonmagnetic) tetragonal phase. }\n\\end{figure}\n\nFinally, the EFGs of the $A$ site doped compounds were calculated with\nVCA. The validity of the VCA was checked by super cell calculations\nfor SrFe$_2$As$_2$ and BaFe$_2$As$_2$. Due to the super cell\nconstruction, there are three different Wyckoff positions for As and\nhence three different EFGs, which lie reasonably close to the VCA EFG\ncurve.  In the VCA calculation we keep the structural parameters fixed\nfor the different levels of doping. In Fig.~\\ref{EFG3} EFGs calculated\nin this manner are shown for CaFe$_2$As$_2$, SrFe$_2$As$_2$ and\nBaFe$_2$As$_2$.  In case of CaFe$_2$As$_2$, the EFG increases when\nelectrons are taken out and decreases when electrons are added to the\nsystem. This implies that the As electron density gets more isotropic,\nwhen the system is electron doped.  For BaFe$_{2}$As$_{2}$, the trend\nis the same as in CaFe$_{2}$As$_{2}$ whereas for SrFe$_2$As$_2$ the\nsituation is slightly different: hole doping does not change the EFG\nmuch, while electron doping increases the EFG. Note however, that the\ncalculated EFG for SrFe$_{2}$As$_{2}$ are quite small.\n\nWe conclude that the EFG in the 122 and in the 1111 systems behave\nsimilar \\cite{Lapaper,Ndpaper}: the effect of electron doping on the\nEFG is much smaller than the influence of the As $z$ position and\npressure (compare Fig.~\\ref{EFG1} and Fig.~\\ref{EFG3}).  This finding\nemphasizes again the crucial importance of a correct description of\nthe Fe-As interaction (that is mostly responsible for the density\naround As) for the physical properties of the iron pnictides.\n\n\\section{{\\bf Results - Experiment}}\n\n\\subsection{{\\bf Substitutions of Fe by other 3$d$-metals}}\n\nAs already mentioned, at ambient pressure, no bulk superconductivity\nhas been observed in stoichiometric $RE$OFeAs and $A$Fe$_2$As$_2$ ($A$\n= Ca, Sr, Ba, Eu) compositions (except for one report on\nSrFe$_2$As$_2$ crystals \\cite{Saha08a}). Instead, these parent\ncompounds display the SDW transition at typically 100--200\\,K. A\nmodification of the intralayer $d^\\mathrm{Fe-Fe}_a$ and interlayer\n$d^\\mathrm{Fe-Fe}_c$ distances and thus of the electronic states at\n$E_F$ can be achieved by different means: (i) application of\nhydrostatic (or uniaxial) pressure, (ii) isovalent substitution of a\nconstituent atom by a smaller\/larger ion in order to apply chemical\npressure, (iii) hole doping or (iv) electron doping by non-isovalent\nsubstitution of any of the constituent atoms. The latter two methods\nusually also excert chemical pressure. Most excitingly, all these\nmethods have been proven to be successful in generating bulk\nsuperconductivity in iron arsenide systems.\n\nApplication of pressure has widely been used to explore the phase\ndiagrams of the superconducting chemical systems\n\\cite{Torikachvili08a,Park08a,Alireza08a}. The method usually introduces\nno crystallographic disorder in the structural building units. In\ncontrast when\nsubstituting a chemical constituent (methods ii--iv) always a certain\ndegree of structural disorder is introduced. First, only substitutions\non sites \\textit{in-between the Fe-As layers} were attempted. Based on\nthe experience gained from extensive work on cuprates, such an indirect\ndoping of the Fe-As layers is expected to introduce only minor\nstructural disorder. In a localized (cuprate-like) as well as in an\nitinerant model of the arsenides, this type of doping amounts to a\nsimple charge doping. Such experiments are therefore not suitable for\ndiscriminating between both models.\n\nIn contrast, a substitution of an atom species \\textit{within the\n  Fe-As layer} can yield more information on the underlying\nphysics. In an itinerant model the substitution of a small amount of\nFe by another $d$ element ($TM$) is expected to be similar to indirect\ndoping since only the total count of electrons is relevant, i.e.\\ a\nrigid-band picture should work in first approximation. In a picture\nwith localized $d$ electrons, on the other hand, doping on the Fe site\nshould directly affect the correlations in the Fe-As layers. A\nbehaviour drastically different from indirect doping should evolve. In\ncuprates the substitution of a few percent Ni or Zn on the Cu site\nleads to a strong reduction of $T_c$.\n\nTherefore, several groups recently investigated the properties of\nsolid solutions of the type $RE$O(Fe$_{1-x}$Co$_x$)As or\n$A$Fe$_{2-x}$Co$_x$As$_2$. Sefat \\textit{et al.}\\,\\cite{Sefat08aetal}\nand Wang \\textit{et al.}\\,\\cite{CWang08aetal} first reported\nsuperconductivity in cobalt doped LaOFeAs with a maximum $T_c \\approx\n10$\\,K. Our group concentrated on the system SrFe$_{2-x}TM_x$As$_2$:\nwhile the pure Fe compound undergoes a lattice distortion and SDW\nordering at $T_0$ = 205\\,K \\cite{Krellner08a}, Co substitution leads\nto a rapid decrease of $T_0$, followed by the onset of bulk\nsuperconductivity in the concentration range $0.2 \\leq x \\leq 0.4$\n\\cite{LeitheJasper08b}. The maximum $T_c$ of $\\approx 20$\\,K is\nachieved for $x \\approx 0.20$. This was in fact also the first\nobservation of \\textit{electron-doping} induced superconductivity in\n$A$Fe$_2$As$_2$ compounds. Co substitution also generated bulk\nsuperconductivity with maximum $T_c \\approx 22$\\,K in\nBaFe$_{2-x}$Co$_x$As$_2$ \\cite{Sefat08b}, however, the optimal doping\nseems to be lower than in the Sr system\n\\cite{Tanatar08a}. Substitution of the following $TM$, nickel,\nintroducing twice as many electrons per atom into the Fe-As layer,\nalso generates bulk superconductivity, albeit with lower $T_c$ than Co\nsubstitution in the Sr compound \\cite{LeitheJasper08b}. However, for\nthe corresponding Ba compound $T_c$ up to 21\\,K is reported for\nBaFe$_{1.90}$Ni$_{0.10}$As$_2$ \\cite{LJLi08a}. Only very recently\nanother internal substitution, namely of As by P, was reported for\nEuFe$_2$As$_2$ \\cite{ZRen08a} and LaOFeAs \\cite{CWang08c}. Also by\nthis means the SDW transition can be influenced and superconductivity\ncan be induced.\n\nIn Table \\ref{tableTc} we present the lattice parameters and the SDW\nand superconducting transition temperatures ($T_0$, $T_c$) of several\nSrFe$_{2-x}TM_x$As$_2$ solid solutions. Values for $T_0$ can be most\neasily obtained from the corresponding anomaly in resistivity data\n(see Fig.\\ \\ref{figrho}). Besides Co and Ni substitutions in the Sr122\nand Ba122 systems, no further $d$ element substitutions have been\nreported yet. As demonstrated recently \\cite{LeitheJasper08b},\nsubstitution of Fe by Co suppresses rapidly $T_0$ (see\nFig.\\ \\ref{figrho}). Bulk superconductivity, as proven by specific\nheat, magnetic shielding, and resistivity data, appears when $T_0 = 0$\nor $T_0 < T_c$, which is reached for $x > 0.20$\n\\cite{LeitheJasper08b}. Only about half of the substituting element\n($x \\approx 0.10$) is necessary to induce bulk superconductivity when\nusing nickel \\cite{LJLi08a}. Both elements then introduce 0.2 excess\nelectrons into the FeAs layers. While the $a$ lattice parameter does\nnot change significantly with Co substitution the $c$ lattice\nparameter decreases continuously in SrFe$_{2-x}$Co$_x$As$_2$\n\\cite{LeitheJasper08b} and with Ni content in BaFe$_{2-x}$Ni$_x$As$_2$\n\\cite{LJLi08a}. Chemical homogeneity of the Co or Ni distribution is\nstill an issue in current samples. One of the most important\nquestions, the co-existence or mutual exclusion of SDW state and bulk\nsuperconductivity, bas been discussed heavily for\n$A$Fe$_2$As$_2$-based alloy series\n\\cite{Rotter08c,JHChu08a,XFWang08a}. The answer is currently open and\ncan be only given for really homogeneous samples.\n\nDirect, in-plane \\textit{hole} doping might also induce\nsuperconductivity. Our new investigations show, that a substitution of\nFe by Mn is possible and that it leads to a continuous increase of\nboth the $a$ and $c$ lattice parameters with Mn content. Under these\nconditions a hole doping does not generate superconductivity. In\ncontrast, in the indirectly-doped Sr$_{1-x}$K$_x$Fe$_2$As$_2$\n\\cite{Sasmal08a,GFChen08aetal} and Ba$_{1-x}$K$_x$Fe$_2$As$_2$\n\\cite{Rotter08b} compounds the $a$ lattice parameter decreases with\n$x$ while $c$ increases, keeping the unit cell volume almost\nconstant. Also, the SDW transition temperature $T_0$ is suppressed\nwith increasing Mn content in a different way (see Fig.\\ \\ref{figrho})\nthan for the Co and Ni substitutions where $T_0$ is suppressed for $x\n\\approx 0.20$ and $x \\approx 0.10$, respectively. Indeed, after an\ninitial decrease of $T_0$ to $\\approx$140\\,K for $x = 0.20$ the\ntransition temperature seems not to decrease further with $x$. Thus,\nthe doping with holes or electrons is not the only important factor\nfor the appearance of superconductivity but a corresponding tuning of\nthe distances $d^\\mathrm{Fe-Fe}_a$ and $d^\\mathrm{Fe-Fe}_c$ has to be\naccomplished also. At present, the microscopic origin of the\ndifferences upon Mn doping compared to Co or Ni doping is unclear.\nTherefore, further investigations for different substitutions are\ncurrently underway.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[height=0.8\\textwidth,angle=90]{KasinathanFigrho.ps}\n\\caption{\nElectrical resistivity of polycrystalline\nSrFe$_{2-x}TM_x$As$_2$ ($TM$ = Mn, Co, Ni) samples\n\\label{figrho}}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Lattice parameters $a,c$ of some SrFe$_{2-x}TM_x$As$_2$\n  (nominal compositions) phase and superconducting transition\n  temperature $T_c^\\mathrm{mag}$ from magnetization measurements\n  (onset, $T_\\mathrm{min} = $1.8\\,K).\n\\label{tableTc}}\n\\begin{tabular}{lcccccl}\n$TM$& $x$  & $a$        & $c$        &$T_c^\\mathrm{mag}$& $T_0$     & Ref.                   \\\\\n    &      & (\\AA)      & (\\AA)      & (K)              & (K)       &                        \\\\ \\hline\n--  & 0.00 & 3.924(3)   & 12.38(1)   & --               & 205       & \\cite{Krellner08a}     \\\\ \\hline\nCo  & 0.10 & 3.9291(1)  & 12.3321(7) & --               & 130       & \\cite{LeitheJasper08b} \\\\\n    & 0.15 & 3.9272(1)  & 12.3123(5) & --               &  90       & \\cite{LeitheJasper08b} \\\\\n    & 0.20 & 3.9278(2)  & 12.3026(2) & 19.2             & \\textless~30     & \\cite{LeitheJasper08b} \\\\\n    & 0.25 & 3.9296(2)  & 12.2925(9) & 18.1             & --        & \\cite{LeitheJasper08b} \\\\\n    & 0.30 & 3.9291(2)  & 12.2704(8) & 13.2             & --        & \\cite{LeitheJasper08b} \\\\\n    & 0.40 & 3.9293(1)  & 12.2711(7) & 12.9             & --        & \\cite{LeitheJasper08b} \\\\\n    & 0.50 & 3.9287(2)  & 12.2187(9) & --               & --        & \\cite{LeitheJasper08b} \\\\\n    & 2.00 & 3.9618(1)  & 11.6378(6) & --               & --        & \\cite{LeitheJasper08b} \\\\ \\hline\nNi  & 0.10 & 3.9299(1)  & 12.3238(6) & $\\approx 8$      & \\textless~85      & \\cite{LeitheJasper08b} \\\\ \\hline\nMn  & 0.10 & 3.9319(2)  & 12.4161(7) & --               & 165       & this work              \\\\\n    & 0.20 & 3.9384(3)  & 12.4615(23)& --               & 130       & this work              \\\\\n    & 0.30 & 3.9441(2)  & 12.4832(7) & --               & 130       & this work              \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\noindent\n\nMagnetic susceptibility and specific heat data for polycrystalline\nsamples SrFe$_{2-x}$Co$_x$As$_2$ have already been presented\n\\cite{LeitheJasper08b}. Here, instead we report newer data obtained on\ntwo crystals with Co contents $x > 0.2$ grown by a flux method (see\nsection \\ref{experimental}).\n\nBoth crystals show strong diamagnetic signals in measurements after\nzero field cooling (ZFC). The onset temperatures $T_c^\\mathrm{mag}$\nare 17.8 K and 15.3\\,K, respectively. While the transition for crystal\nX1 is much wider than that of crystal X2 the $T_c$ of the latter is\nsomewhat lower, indicating a slightly larger Co-content in accordance\nwith the EPMA investigations. The shielding signals (ZFC) corresponds\nto the whole sample volume, however the Meissner effect (FC) is very\nsmall. This is typical for Co-substituted $A$Fe$_2$As$_2$ materials\nand probably due to strong flux line pinning. The random (and somewhat\ninhomogeneous) substitution of Fe by Co within the superconducting\nlayers seems to introduce effective pinning centers. This is a\nremarkable difference to superconducting compositions with\nsubstitutions outside the Fe-As layers.\n\nAs already demonstrated \\cite{LeitheJasper08b}, the SDW ordering and\nthe connected lattice distortion at $T_0$ = 205\\,K in SrFe$_2$As$_2$\n\\cite{Krellner08a,Tegel08a} is strongly suppressed by Co substitution,\nsimilar as for K substitution (indirect hole doping)\n\\cite{GFChen08aetal}. For the two crystals no corresponding anomaly in\n$\\rho(T)$ is observed.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[height=0.8\\textwidth,angle=90]{KasinathanFigCp.ps}\n\\caption{Molar isobaric specific heat $c_p\/T^{2}$ of\nSrFe$_{2-x}$Co$_x$As$_2$ crystal samples for different magnetic fields.\nFor crystal X1 only data for $\\mu_0H = 0$ are given. Zero-field data\npoints are connected by a line. The inset shows the resistivity of\ncrystal X2 around $T_c$ for $\\mu_0H$ = 0.01, 3, 6, and 9\\,T.\n\\label{figcp}}\n\\end{center}\n\\end{figure}\n\nThe specific heat $c_p(T)$ for the two crystals are shown in\nFig.\\ \\ref{figcp} in a $c_p(T)\/T^{2}$ versus $T$ representation. It\ncan be clearly seen that crystal X1 has a very broad transition (with\na ``foot'' at the high-temperature side) while crystal X2 displays a\nrounded but single-step anomaly. The specific heat jumps $\\Delta\nc_p\/T_c$ and the transition temperatures $T_c^\\mathrm{cal}$ can be\nevaluated by a fit including a phonon background (harmonic lattice\napproximation) and an electronic term $c_{es}(T)$ according to the BCS\ntheory ($\\Delta c_p\/T_c = 1.43\\gamma$) or the phenomenological\ntwo-liquid model ($\\Delta c_p\/T_c = 2\\gamma$, a model for stronger\ne-ph coupling). The inclusion of a residual linear term $\\gamma'T$ was\nfound to be absolutely necessary for a good fit.\n$$\nc_p(T) = \\gamma' \\: T + \\beta T^{3} + \\delta T^{5} + c_{es}(T)\n$$\nThe jump at $T_c$ is ``broadened'' in order to simulate the\nrounding of the transition steps due to chemical inhomogeneities. For\nthe more homogeneous sample X2 no difference in the least-squares\ndeviation is observed between the BCS and two-fluid model. We find for\ncrystal X2 $\\Delta c_p\/T_c^\\mathrm{cal} \\approx$12.0 mJ mol$^{-1}$\nK$^{-2}$ at $T_c^\\mathrm{cal}$ = 12.34\\,K and also for crystal X1\nsimilar values ($\\approx$12.5 mJ mol$^{-1}$ K$^{-2}$; $T_c$ = 13.22\\,K).\nSimilar values of $\\Delta c_p\/T_c^\\mathrm{cal}$ were observed in our\nprevious study \\cite{LeitheJasper08b} of polycrystalline samples with\nCo-contents $x = 0.20$ and $x = 0.30$.\n\nThe existence of a linear specific heat term well below $T_c$ is found\nfor several $A$Fe$_2$As$_2$-based alloys\n\\cite{LeitheJasper08b,GMu08a}.  For $H = 0$ we observe for both\ncrystals values of $\\gamma'$ around 20 mJ mol$^{-1}$\nK$^{-2}$. $\\gamma'$ generally increases with field.  Whether the\nresidual $\\gamma'$ is due to defects as in the case of early cuprate\nsuperconductor samples (see e.g. Ref.\\,\\cite{TrisconeJunod96}) or\nwhether it is an intrinsic contribution has to be clarified by further\nexperiments. An intrinsic reason could be some ungapped parts of the\nFermi surface \\cite{Drechsler03aetal}. For the specific heat jump\n$\\Delta c_p\/T_c$ in BaFe$_{2-x}$Co$_x$As$_2$ also relatively small\nvalues are reported ($\\approx$ 25 mJ mol$^{-1}$ K$^{-2}$\n\\cite{Tanatar08a}) while for Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ ($\\Delta\nc_{p}\/T_c \\approx$ 100 mJ mol$^{-1}$ K$^{-2}$ \\cite{GMu08a,Rotter08c})\nmuch larger jumps are observed. This may indicate that the\nsuperconducting Fermi surface portions in the Co-substituted compounds\n(in-plane doping) are strongly different from those in indirectly\ndoped superconductors.  Recent photoemission (ARPES) investigations\nindeed point out severe differences between (non-superconducting)\nBaFe$_{1.7}$Co$_{0.3}$As$_2$ and (superconducting)\nBa$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ \\cite{Sekiba08a}.  In conclusion, our\ndoping experiments on the Fe site strongly favor an intinerant picture\nover a localized scenario.  Further thermodynamic and Fermi surface\nstudies are required to resolve this issue.\n\n\n\n\\section{\\bf Summary}\n\nIn this paper, we presented a joint theoretical and experimental study\nof the systems $A$Fe$_{2}$As$_{2}$ ($A$ = Ca, Sr, Ba, Eu) and\nSrFe$_{2-x}$$TM_{x}$As$_{2}$ ($TM$ = Mn, Co, Ni) to investigate the\nrelation of crystal structure and charge doping to magnetism and\nsuperconductivity in these compounds.  Based on {\\it ab-initio}\nelectronic structure calculations we focused on the relationship\nbetween the crystal and electronic structure, charge doping and\nmagnetism for the 122 family since their electronic structure and\nphysical properties are quite similar  to the other superconducting\niron pnictide families (1111, 1111', 111, 11).\n\nAlthough problems with an accurate description of the Fe-As\ninteraction persist in present-day density functional calculations,\nthis approach provides deep inside into many questions.  We\ndemonstrated that tetragonal to orthorhombic transition in the 122\ncompounds is intrinsically linked to the SDW formation in agreement\nwith experimental observations. We find an anisotropic,\npressure-induced volume collapse for $A$Fe$_{2}$As$_{2}$ ($A$ = Ca,\nSr, Ba, Eu) that goes along with the suppression of the SDW magnetic\norder. For $A$ = Ca our calculations are in excellent agreement with\nthe experimental observations \\cite{Kreyssig08a}. An experimental\nverification for the other compounds ($A$ = Ca, Sr, Ba, Eu) would be\ndesirable. With respect to the doping dependence in\nSrFe$_{2-x}$$TM_{x}$As$_{2}$, we find the correct trends compared to\nthe experimental results. A more quantitative comparison will require\nthe explicit treatment of the influence of the substitutional disorder\non electronic structure and magnetism. This task is left for an\nextended future study.\n\nAs demonstrated also for the EFG, many properties of these compounds\nare sensitive to the As $z$ position. Since the magnetism and\ntherefore the superconductivity crucially depend on this structural\nfeature and the related accurate description of the Fe-As interaction,\nthe improvement of the calculations in this respect may offer the key\nto the understanding of superconductivity in the whole family. In\norder to improve the present-day density functional calculation for\nthis class of materials, the first step requires a deeper\nunderstanding of where, how and why these DFT calculation\nfail. Besides further calculational effort, a broader experimental\nbasis, especially high pressure studies will be necessary to approach\nthis complex issue.\n\nExperimentally, we investigate the substitution of Fe in\nSrFe$_{2-x}TM_{x}$As$_{2}$ by other 3$d$ transition metals, $TM$ = Mn,\nCo, Ni.  In contrast to a partial substitution of Fe by Co or Ni\n(electron doping) a corresponding Mn partial substitution does not\nlead to the suppression of the antiferromagnetic order or the\nappearance of superconductivity.  \n\nThe observed existence of a linear specific heat term in\nSrFe$_{2-x}$Co$_{x}$As$_{2}$ well below $T_c$ is extremely important for\nthe understanding of the superconductivity in this compound and all\nrelated 122 superconductors with doping on the Fe site. Therefore, a\ncareful investigation whether this feature is intrinsic or not is a\ncrucial question. In order to answer it, further studies on carefully\nprepared and characterized high-quality samples are required. Since in\nmany experiments a considerable sample dependence is observed, this\nissue is of large general importance for a future understanding of the\nsuperconducting iron pnictide materials.\n\nWe thank T.\\ Vogel, R.\\ Koban, K.\\ Kreutziger, Yu.\\ Prots, and R.\\\nGumeniuk for assistance.\n\n\n\n\n\n\\providecommand{\\newblock}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe increasing penetration of renewable energies challenges the conventional power system operation paradigm, in particular, the economic dispatch (ED) process. Conventionally, the short term load prediction can be rather accurate, and hence, the dispatch based on those predictions yields near minimal generation cost. However, when renewable generations are considered as negative loads, their stochastic nature makes the net load hard to predict. To improve the effectiveness of ED, various advanced load forecasting methods have been proposed over the past two decades.\n\nNonetheless, more accurate load forecasting does not necessarily lower the generation cost. This is largely because the common forecasting precision metric is the mean square error (MSE) over the period of interest, and this common metric usually won't coincide with the objective function of ED.\n\nThis motivates us to adopt the notion of end-to-end machine learning and propose a task specific criteria to conduct load forecasting. However, to design an effective framework is delicate since end-to-end machine learning usually suffers from low data utilization\\cite{b}. To tackle this challenge, we propose an efficient optimization kernel to speed up the training process, which improves the data utilization. The optimization kernel further motivates us to design a robust model-free end-to-end learning framework.\n\n\\subsection{Related Works}\nWe identify two major bodies of closely related research directions in designing the learning framework for ED: one seeks to provide effective load forecasting methods while the other investigates various ways to conduct efficient ED.\n\nLoad forecasting is a rather classical technique in power system operation, and has been well investigated (see \\cite{r} for a comprehensive survey). While most classical methods for load forecasting utilize the statistical analytics (e.g., the adaptive autoregressive moving-average (ARMA) model in \\cite{l1}) or time series analysis (e.g., stochastic time series analysis in \\cite{l3}), machine learning algorithms have also been applied to load forecasting since the mid-1980s \\cite{lf1}. With the advance in machine learning over the last decade, this line of research attracts significant attention.\nJust to name a few, Lee \\emph{et al}. design a neural network based model to capture weekend-day energy consumption patterns in \\cite{l4}. Bashir \\emph{et al}. seek to improve the effectiveness of neural network by Particle Swarm Optimization (PSO) in \\cite{l5}.\nJaved \\emph{et al.} combine the Artificial Neural Network (ANN) and Support Vector Machine (SVM) for load forecasting in \\cite{l7}.\nWe want to emphasize that the conventional forecasting precision metric in the literature is MSE. In our work, we aim to highlight the mismatch between MSE and the desirable task specific criteria.\n\n\n\n\n\n\nEconomic dispatch is one of the most important processes in power system operation. There is a huge body of related literature to improve the effectiveness of ED. The major difficulties come from the temporal coupling and the dynamic implementation.\nTo tackle temporal challenges, the solution concepts range from classical linear and quadratic programming \\cite{b3} to genetic algorithm for value point loading \\cite{b4}. To overcome the difficulties of dynamic dispatch, various mathematical programming approaches have been proposed, including Lambda iterative method \\cite{b7}, interior point method \\cite{b8,b9}, and dynamic programming \\cite{b10}. However, mathematical programming approaches are usually time consuming and not ideal for large scale systems. Hence, for large scale non-convex ED, heuristic and hybrid methods are often preferable, e.g., hybrid evolution programming in \\cite{b11}.\n\n\n\n\nHowever, the notion of using learning framework to conduct ED only appears recently.\nDonti \\emph{et al}. propose a generic end-to-end machine learning framework for stochastic programming, with an application to the single generator ED problem in \\cite{z}. Different from this work, we target to design an end-to-end machine learning framework for general ED.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Our Contributions}\nIn seek of designing the effective end-to-end machine learning framework, our principal contributions can be summarized as follows.\n\n\n\n\n\\begin{itemize}\n    \\item \\emph{Task Specific Criteria}:  We identify that MSE is not ideal to evaluate the performance of load forecasting for conducting ED. We adopt the notion of end-to-end machine learning to derive the task specific criteria.\n    \\item \\emph{Optimization Kernel Construction}: To avoid solving the multivariate constrained stochastic optimization problem during each learning iteration, we exploit the problem structure and propose an efficient way to construct the optimization kernel for effective learning.\n    {\n    \\item \\emph{Model-Free End-to-End Learning}: Motivated by the optimization kernel, we propose a model-free approach to further improve the efficiency and effectiveness of our end-to-end learning framework.}\n  \n \n\\end{itemize}\n\nThe rest of the paper is organized as follows. \nSection \\ref{formulation} presents the system model. We revisit the conventional learning framework for ED in Section \\ref{conventional_frame}. Then, in Section \\ref{e2elearning_basis}, we lay out the theoretical foundation for the end-to-end learning framework for ED, which can be time consuming. To tackle the challenge, we design the optimization kernel in Section \\ref{implementation}, and then propose the model-free framework in Section \\ref{modelfree}.\nNumerical studies verify the efficiency and effectiveness of our proposed frameworks in Section \\ref{experiment}. Finally, concluding remarks and future directions are given in Section \\ref{conclude}. We defer all the necessary proofs in the Appendix.\n\n\\section{Problem Formulation}\n\\label{formulation}\nWe consider a standard ED process over a period of $T$ time slots with geographically distributed $n$ generators and $m$ demands. Besides the generation cost for each generator, we assume there are certain risk costs associated with the supply demand mismatch in real time. To capture the stochastic nature of net demands, they are modeled as random variables, following possibly distinct distribution. The system operator seeks to minimize the total cost of the system by solving the following optimization problem :\n\\begin{align}\n    \\min \\quad &\\sum_{t=1}^{T}{\\left(\\sum_{i=1}^{n}c_i({g}_{it})+\\gamma_1{\\mathbb{E}(S_t)}^++\\gamma_2{\\mathbb{E}(-S_t)}^+\\right)}\n    \\label{eq1}\\\\\n    s.t. \\quad \n    &0 \\leq g_{it} \\leq B_i, \\quad \\forall{i,t}\n    \\label{eq2}\\\\\n    &-\\boldsymbol{b} \\leq \\mathbb{E}(H_g\\boldsymbol{g}_{t}-H_d\\boldsymbol{\\hat{\\boldsymbol{d}}_t}) \\leq \\boldsymbol{b}, \\quad\n    \\forall{t}\n    \\label{eq3}\\\\\n    &S_{t} = \\sum \\nolimits_{j=1}^m{\\hat{d}_{jt}}-\\sum\\nolimits_{i=1}^n{g_{it}},\n     \\quad \\forall{t}.\n     \\label{eq4}\n\\end{align}\n\nThe decision variables are $g_{it}$'s, and each $g_{it}$ denotes the dispatched generation of generator $i$ at time $t$.  We denote $\\boldsymbol{g}_t$ the vector of $(g_{1t},...,g_{nt})$. On the other hand, the parameters in the optimization problem are defined as follows.\n\\begin{itemize}\n    \\item $\\hat{d}_{jt}$ : prediction of demand $j$ at time $t$, $\\hat{\\boldsymbol{d}_t} = (\\hat{d}_{1t},...,\\hat{d}_{mt})$.\n    \\item $\\gamma_1$, $\\gamma_2$ : unit generation shortage and excess penalties\n    \\item $S_t$ : total shortage based on predicted demands at time $t$\n    \\item $B_i$ : generation capacity of generator $i$\n    \\item $\\boldsymbol{b}$ : transmission line capacity vector\n    \\item $H_g$, $H_d$: generation and load shift factor matrices\n    \\item $c_i(\\cdot)$ : generation cost function of generator $i$\n\\end{itemize}\n\nConstraint (\\ref{eq2}) captures the limited capacity of each generator. Constraint (\\ref{eq3}) uses the shift factor matrices \\cite{mp} to represent the DC approximation of line capacity constraints.\nWe follow the literature and assume the generation cost functions (i.e., $c_i(\\cdot)$'s) are linear for each individual generator. Hence, the total individual cost function is piecewise linear.\nNote that, the randomness in the predicted demands ($\\hat{d}_{jt}$'s) makes the optimization problem (\\ref{eq1})-(\\ref{eq4}) a multivariate stochastic optimization with linear constraints. \n\n\n\n\n\n\n\n\n\n\\section{Conventional Learning Framework for ED}\n\\label{conventional_frame}\nIn this section, we first introduce the conventional learning framework for ED and then use the electricity pool model to highlight its drawback: the mismatch between MSE learning criteria and the ultimate goal of minimizing system cost.\n\\subsection{Conventional Framework}\nConventional wisdom isolates the load prediction from the whole ED process and aims to obtain a perfect load predictor across time. Hence, the major task of conventional learning framework for ED is to train an accurate predictor. And the ED process will directly take the predicted loads as single value inputs and conduct the dispatch. The learning process is visualized in Fig.\\ref{fig1}.\n\\begin{figure}[t]\n    \\centerline{\\includegraphics[width=8cm]{Figure2.pdf}}\n    \\centering\n\\caption{Conventional Learning Process.}\n\\label{fig1}\n\\end{figure}\nThe whole process is intuitive and easy to understand. \nHowever, the training criteria (MSE) is a generic selection, which is not customized for ED. Hence, the load predictor trained through this process does not necessarily lower the generation cost in practice.\n\n\n\n\\subsection{Inefficiency of MSE based Predictor}\n\nTo highlight the fact that MSE can be inefficient, we consider the ED in the electricity pool model, where all the network constraints are ignored and the system cost function degenerates to a single piecewise linear function.\n\nMathematically, we simplify the optimization problem (\\ref{eq1})-(\\ref{eq4}) as follows:\n\\begin{equation}\n    \\begin{split}\n         \\min \\quad &\\sum\\nolimits_{t=1}^{T}\\left[\\hat{C}_t({g}_{t})+\\gamma_1{\\mathbb{E}(S_t)}^++\\gamma_2{\\mathbb{E}(-S_t)}^+\\right]\\\\\n    s.t. \\quad&S_{t} = \\sum \\nolimits_{j=1}^m{\\hat{d}_{jt}}-\\sum\\nolimits_{i=1}^n{g_{it}},\n     \\quad \\forall{t},\n    \\end{split}\n    \\label{elecpool}\n\\end{equation}\nwhere for each $t$,\n\\begin{equation}\n    \\begin{split}\n        \\hat{C}_t(g_t) = \\min\\quad&\\sum\\nolimits_{i=1}^n{c_i(g_{it})}\\\\\n    s.t.\\quad&0 \\leq g_{t} := \\sum\\nolimits_{i=1}^n{g_{it}} \\leq \\sum\\nolimits_{i=1}^n B_i.\n    \\end{split}\n\\end{equation}\n\nNote that, in optimization problem (\\ref{elecpool}), we only need to consider the total dispatched generation $g_t$'s, instead of $g_{it}$'s, as it is easy to recover the vector $(g_{it}, \\forall{i})$ from $g_t$ based on the merit order of the generator's marginal cost. It is also clear that in this simplified model, the dispatch decisions $g_t$'s are decoupled across time. To capture the role of load predictor in the decision making, we define\n\\begin{align}\n    \\hat{d}_t = \\sum\\nolimits_{j=1}^m{\\hat{d}_{jt}} . \n\\end{align}\n\nWe further denote $f_t(x)$ and $F_t(x)$ the probability density function (\\emph{pdf}) and the cumulative density function (\\emph{cdf}) of $\\hat{d}_t$. They allow us to express the risk cost $\\hat{R}(g_t,f_t(x))$ as follows.\n\\begin{equation}\n\\begin{aligned}\n    \\hat{R}(g_t,f_t(x))=& \\gamma_1\\int_{g_t}^{\\infty}{(x-g_t)f_t(x)dx} \\\\\n    &+ \\gamma_2\\int_{-\\infty}^{g_t}{(g_t-x)f_t(x)dx}.\n    \\end{aligned}\n\\end{equation}\nBased on $\\hat{R}(g_t,f_t(x))$, we can rewrite the decision making problem for each time $t$:\n\\begin{equation}\n    \\begin{split}\n        \\min \\quad & \\hat{C}_t({g}_{t})+ \\hat{R}(g_t,f_t(x)) \\\\\n    s.t. \\quad &0 \\leq g_{t} \\leq \\sum\\nolimits_{i=1}^n B_i.\n    \\end{split}\n\\end{equation}\n\nWe assume the generation shortage penalty is larger than the marginal generation cost, and far overweigh the excessive penalty, i.e., $\\gamma_1 \\!>\\! C'(g_t)$, $\\gamma_1 \\gg \\gamma_2$.\nThis is reasonable as $\\gamma_1$ represents the marginal generation cost for the next more expensive generator while $\\gamma_2$ can be understood as the opportunity cost for the undispatched generators.\nThe first order optimality condition indicates the optimal dispatch should satisfy the following condition:\n\\begin{align}\n     g_t = \\min \\left\\{F_t^{-1}\\left(\\frac{\\gamma_1-\\hat{C}_t'(g_t)}{\\gamma_1+\\gamma_2}\\right), \\sum\\nolimits_{i=1}^n{B_i}\\right\\}.\n\\end{align}\n\nUnder normal operating conditions, the available generation ($\\sum\\nolimits_{i=1}^n{B_i}$) is always greater than the peak demand. Hence, we can safely drop the min operator in practice:\n\\begin{align}\n    g_t = F_t^{-1}\\left(\\frac{\\gamma_1-\\hat{C}_t'(g_t)}{\\gamma_1+\\gamma_2}\\right).\n\\end{align}\n\nFollowing the notion that generation and the predicted demand should be aligned, this notion indicates that if we are only allowed to obtain a single valued load prediction, we should deliver $d^*_t$ to the system operator, where\n\\begin{align}\\label{desired_criteria}\n    d_t^* = F_t^{-1}\\left(\\frac{\\gamma_1-\\hat{C}_t'(d_t^*)}{\\gamma_1+\\gamma_2}\\right).\n\\end{align}\n\nOn the other hand, the learning framework based on MSE criteria will deliver $d_t^* = \\mathbb{E}[\\hat{d}_t]$ to the system operator, which is often different from the percentile\\footnote{Although some precision metrics like mean absolute error (MAE) will lead to a percentile form for $d^*_t$, they are often independent of $\\gamma_1$, $\\gamma_2$, and the cost functions. Hence, they are also fundamentally different from task specific criteria.} represented in Eq. (\\ref{desired_criteria}). Hence, the conventional framework is not efficient in minimizing the system cost even in the simplified electricity pool model.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{End-to-End Learning : the Basis}\n\\label{e2elearning_basis}\n\n\n\n\nEnd-to-end learning is a powerful tool in the machine learning community. In this section, we revisit the basic concepts in the generic end-to-end learning framework, and then introduce the notion of task specific criteria for ED.\n\n\nSpecifically, as for learning the ED policies, end-to-end learning would directly learn the final dispatch policies given the input data and the loss function for training, which often measures the difference between predicted policy $\\hat{\\boldsymbol{g}}_{t}$ and the optimal policy ${\\boldsymbol{g}}^*_{t}$, i.e, $\\sum\\nolimits_{t = 1}^{T}\\sum\\nolimits_{i = 1}^n|\\hat{{g}}_{it} - g^*_{it}|$.\n\nHowever, ignoring all the intermediate stages makes pure end-to-end learning suffer from a number of disadvantages.\nFirst, the predicted dispatch policy may not satisfy all the constraints in ED. Also, this framework can be rather data inefficient, especially for large dynamic system control, such as ED in power system. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Task Specific Criteria for ED}\nIn order to avoid the shortcomings of pure end-to-end learning, \na task specific criteria can be helpful.\nSpecifically,\nwe can carefully examine the structure of problem (\\ref{eq1})-(\\ref{eq4}).\nGiven the probability distribution of all the random variables, we can convert this stochastic optimization into a deterministic optimization.\n\nNote that, since our formulation ignores the ramping constraints, the decision makings across time are naturally decoupled. Such simplifications allow us to highlight the effectiveness of task specific end-to-end machine learning. \nHence, in the subsequent analysis,\nwe will focus on single shot ED and provide our insights on how to generalize our approach to ED with ramping constraints by the end of this section. Suppose we are given the distribution of $\\hat{d}_t$, then the single time shot ED degenerates to the following problem:\n\\begin{equation}\n\\begin{split}\n    \\min \\quad &\\sum_{i=1}^{n}c_i({g}_{it})+\\gamma_1{\\mathbb{E}(\\hat{d}_t-g_t)}^++\\gamma_2{\\mathbb{E}(g_t-\\hat{d}_t)}^+\\\\\n    s.t. \\quad \n    &0 \\leq g_{it} \\leq B_i, \\quad \\forall{i}\\\\\n    &-\\boldsymbol{b} \\leq \\mathbb{E}(H_g\\boldsymbol{g}_{t}-H_d\\boldsymbol{\\hat{\\boldsymbol{d}}_t}) \\leq \\boldsymbol{b}. \n\\end{split}\n\\label{ED_tsk}\n\\end{equation}\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Lemma 1}:\nThe single shot ED problem (\\ref{ED_tsk}) is convex.\n\\vspace{0.1cm}\n\nThe proof can be immediately obtained by examining the Hessian matrix of the objective function with respect to ${g_{it}}$'s and showing it is positive semi-definite.\nThe objective function in (\\ref{ED_tsk}) can be used to derive the chain rule for back propagation in the task specific end-to-end learning.\n\nSuppose $\\hat{d}_t$ follows some distribution $f_t(x;\\boldsymbol{\\hat{\\theta}}_t)$, where $\\boldsymbol{\\hat{\\theta}}_t$ represents the parameters of the distribution (e.g., parameter $\\lambda$ for Exponential distribution). Then, the goal of the task specific predictor is to estimate the distribution $f_t$ with the appropriate parameters $\\boldsymbol{\\hat{\\theta}}_t$. Given the predicted distribution, we define the task loss function by $L_t(\\boldsymbol{\\hat{g}}_t,d_t)$:\n\\begin{equation}\n\\!L_t(\\boldsymbol{\\hat{g}}_t,d_t) \\!=\\! C(\\boldsymbol{\\hat{g}}_t)\\!+\\gamma_1{({d}_t-\\hat{g}_t)^+}\\!+\\gamma_2(\\hat{g}_t-{d}_t)^+\\!-C_t(d_t),\n\\end{equation}\nwhere $d_t$ denotes the true load.\nThe other notations $C(\\boldsymbol{g}_t)$, $C_t(d_t)$, and $\\hat{\\boldsymbol{g}}_t$ represent the total generation cost function given dispatch profile $\\boldsymbol{g}_t$, the minimal generation cost to meet the true demand $d_t$, and the optimal generation dispatch profile given estimation $f_t(x;\\hat{\\boldsymbol{\\theta}}_t)$. Formally,\n\\begin{align}\n    C({\\boldsymbol{g}}_t) = \\sum\\nolimits_{i = 1}^n c_i({g}_{it}),\n\\end{align}\n\\begin{equation}\n    \\begin{split}\n        C_t(d_t) := \\min \\quad &\\sum\\nolimits_{i=1}^{n}c_i({g}_{it})\\\\\n    s.t. \\quad&\\text{Constraints }(\\ref{eq2}),(\\ref{eq3})\\\\\n     &\\sum\\nolimits_{i=1}^n g_{it} = d_t,\n    \\end{split}\n    \\label{optkernel}\n\\end{equation}\n\\begin{equation}\n    \\begin{split}\n    \\hat{\\boldsymbol{g}}_t = \\arg\\min \\quad &C(\\boldsymbol{g}_t) +\\hat{R}(g_t,f_t(x;\\hat{\\boldsymbol{\\theta}}_t))\\\\\n    s.t. \\quad &\\text{Constraints }(\\ref{eq2}),(\\ref{eq3}).\n    \\end{split}\n\\end{equation}\n\nNote that, $\\hat{\\boldsymbol{g}}_t$ can be calculated by sequential quadratic programming (SQP). Hence, the chain rule of back propagation aligns with the chain rule for partial derivatives:\n\\begin{align}\n    \\frac{\\partial{L}}{\\partial{\\hat{\\boldsymbol{\\theta}}_t}} = \\frac{\\partial{L_t}}{\\partial{\\hat{\\boldsymbol{g}}_t}}\n    \\cdot\\frac{\\partial{\\hat{\\boldsymbol{g}}_t}}{\\partial\\hat{\\boldsymbol{\\theta}}_t} , \\forall{t}.\n    \\label{chain}\n\\end{align}\n\nThe two partial derivatives on the right hand side in Eq.(\\ref{chain}) can be calculated automatically in the SQP (see \\cite{opt} for more details) without explicit formula.\nWe visualize the whole process in Fig. \\ref{e2e4p}.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=9cm]{e2e4.png}}\n\\caption{Task-Specific Optimization Based Learning Process.}\n\\label{e2e4p}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{End-to-End Learning : Implementation}\n\\label{implementation}\nIn this section, we discuss in detail how to implement the task specific criteria to achieve the best efficiency and effectiveness. Specifically, by observing the structure of the single shot ED problem, we design an optimization kernel to enable efficient learning.\n\n\\subsection{Optimization Kernel for Efficient Learning}\n\nWe first use the notion of parametric analysis to examine problem (\\ref{ED_tsk}).\nThe key parameter in determining the value of the objective function is the same as that for the electricity pool model, the total generation $g_t$. We define $\\tilde{L}_t(g_t)$ to highlight this observation:\n{\n\\begin{equation}\n\\begin{split}\n        \\tilde{L}_t(g_t)\\! :=\\! \\min\\quad&C(\\boldsymbol{g}_t)\\!+\\!\\gamma_1{\\mathbb{E}(\\hat{d}_t\\!-\\!g_t)}^+\\!+\\!\\gamma_2{\\mathbb{E}(g_t\\!-\\!\\hat{d}_t)}^+\\\\\n        s.t.\\quad&\\text{Constraints }(\\ref{eq2}),(\\ref{eq3}),\\\\\n    &g_t = \\sum\\nolimits_{i = 1}^{n}g_{it}.\n\\end{split}\n\\label{Lgt}\n\\end{equation}\n}\n\n$\\tilde{L}_t(g_t)$ can be decomposed into two components. One represents the cost of a fixed structure (the generation cost, $C_t(g_t)$ as defined in Eq.(\\ref{optkernel})) and the other deals with the predicted distribution of $\\hat{d}_t$ (the risk cost, denoted by $\\tilde{R}_t(g_t)$). \nMathematically, \n\\begin{align}\n    \\tilde{L}_t(g_t) = C_t(g_t) + \\tilde{R}_t(g_t),\n\\end{align}\nwhere\n\\begin{align}\n\\tilde{R}_t(g_t) = &\\gamma_1\\mathbb{E}{(\\hat{d_t}-g_t)^+} + \\gamma_2\\mathbb{E}{(g_t-\\hat{d_t})^+}.   \n\\end{align}\n\nDenote the feasible region constructed by constraints of (\\ref{Lgt}) by $\\mathcal{A}(g_t)$.\nDefine\n\\begin{align}\n&g_t^{\\min} = \\inf \\mathcal{A}(g_t),\\\\\n&g_t^{\\max} = \\sup \\mathcal{A}(g_t).\n\\end{align}\n\nThe following lemma indicates the structure of $C_t(g_t)$.\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Lemma 2}: The generation function $C_t(g_t)$ is continuous, piecewise linear, and convex in $g_t$ over $[g_t^{\\min}, g_t^{\\max}]$.\n\\vspace{0.1cm}\n\nInspired by the idea proposed in \\cite{w}, we can efficiently construct $C_t(g_t)$ as described in Algorithm \\ref{CGAs} (Curve Generative Algorithm or CGA in short). \n\n\\begin{algorithm}\n\\SetAlgoLined\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\n\\Input{generation range: $g_t^x$ and $g_t^y$\\;}\n\\Output{function curve of $C_t(g_t)$ over $[g_t^x, g_t^y]$\\;}\nSolve $C_t(g_t^x)$, $C_t(g_t^y)$ and get their Lagrangian multipliers $\\lambda_t^x$, $\\lambda_t^y$, respectively;\n\n\\eIf {$\\lambda_t^x$ == $\\lambda_t^y$}\n{The slope of $C_t(g_t)$ over $[g_t^x, g_t^y]$ is $\\lambda_t^x$;}\n{Solve the equations below to obtain $g_t^z$, $C_z$;\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&C_z - C_t(g_t^x) = \\lambda_t^x(g_t^z-g_t^x)\\\\\n&C_z - C_t(g_t^y) = \\lambda_t^y(g_t^z-g_t^y)\n\\end{aligned}\n\\right.\n\\end{equation}\n}\n\\eIf {$C_z$==$C_t(g_t^z)$}\n{$g_t^z$ is the unique breaking point in $[g_t^x, g_t^y]$, the\nslope over $[g_t^x, g_t^z]$ is $\\lambda_t^x$ and the slope of $[g_t^z, g_t^y]$ is $\\lambda_t^y$;}\n{Recursively generate the curve by \\text{CGA}$(g_t^x,g_t^z)$ and \\text{CGA}$(g_t^z,g_t^y)$;}\nReturn;\n\\caption{CGA($g_t^x$, $g_t^y$)}\n\\label{CGAs}\n\\end{algorithm}\n\n\n\nWith the knowledge of $C_t(g_t)$, the optimal total dispatch $g_t^*$ should satisfy:\n\\begin{equation}\n\\begin{split}\n    g^*_t = \\arg\\min&\\quad \\tilde{L}_t(g_t)\\\\\n    s.t.&\\quad g_t^{\\min} \\leq g_t \\leq g_t^{\\max}.\n\\end{split}\n\\label{g^*_t}\n\\end{equation}\nIt is straightforward to verify that\n\\begin{align}\n    C_t''(g_t) + \\tilde{R}_t''(g_t) > 0, \\quad g_t \\geq 0.\n\\end{align}\n\n{Hence, there must exist a unique $g^*_t$ to problem (\\ref{g^*_t}).}\nWhile $C_t(g_t)$ is not differentiable everywhere over the domain, we can conduct a binary search to efficiently obtain $g^*_t$ for given prediction distributions.\n\nThe only remaining handle is to construct the dispatch policy $\\boldsymbol{g}_t$ from the total dispatch ${g}_t$. Due to the continuity of the solution space of $C_t(g_t)$, we can efficiently construct $\\boldsymbol{g}_t$ with the help of the information embedded in the breaking points obtained in Algorithm \\ref{CGAs}. More precisely, without loss of generality, suppose $g^*_t \\in [g_t^k,g_t^{k+1}]$, where $g_t^k$ and $g_t^{k+1}$ are two adjacent breaking points of $C_t(g_t)$. Denote $\\boldsymbol{g}_t^k$ and $\\boldsymbol{g}_t^{k+1}$ the corresponding dispatch profiles for ${g}_t^k$ and ${g}_t^{k+1}$. Then, continuity of solution space leads to the following lemma:\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Lemma 3}:\nIf $g^*_t = (1-\\gamma)g^k_t + \\gamma{g^{k+1}_t}, 0 \\leq \\gamma \\leq 1$, then the optimal dispatch profile $\\boldsymbol{g}_t^*$ corresponding to the total dispatch $g^*_t$ can be constructed as follows:\n\\begin{align}\n    \\boldsymbol{g}_t^* = (1-\\gamma)\\boldsymbol{g}^k_t + \\gamma{\\boldsymbol{g}^{k+1}_t}.\n\\end{align}\n\\vspace{0.1cm}\n\nThus, we complete the construction of the optimization kernel for the end-to-end framework. We illustrate this process in detail in Algorithm \\ref{kernel}, where we utilize the information of $C'_t(g_t)$. This information can be obtained during the construction process of $C_t(g_t)$. Since $C'(g_t)$ is only well defined over $[g_t^{\\min}, g_t^{\\max}]$, we generalize this derivative beyond $[g_t^{\\min}, g_t^{\\max}]$ as follows:\n\\begin{align}\n  C_t'(g_t)=\\left\\{\n\\begin{aligned}\n&C_t'(g_t^{\\min}),   &g_t \\leq g_t^{\\min} \\\\\n&C_t'(g_t),   &g_t^{\\min} < g_t  < g_t^{\\max} \\\\\n&C_t'(g_t^{\\max}),   &g_t \\geq g_t^{\\max}\n\\end{aligned}\n\\right.  \n\\end{align}\n\\footnotetext[2]{Each piece of data in the two sets can be represented by a pair $(x,d)$, where $x$ denotes the input data, like the previous day's load, week-weekend types and temperatures, while $d$ denote the output data, like the true load of the next day.}\n\\begin{algorithm}\n\\SetAlgoLined\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\n\\Input{training set and validation set $(x,d)$\\footnote[2];}\n\\Output{predictor $P$;}\nCompute $C_t'(g_t)$'s and $\\tilde{R}_t'(g_t)$'s function curve;\n\nCompute $g^{\\min} $ and $g^{\\max} $;\n\nCompute ${G}(g_t)$'s function curve, where $\\boldsymbol{g}_t = G(g_t)$;\n\n\\While{Loss of validation set $L_v$ doesn't increase}\n{\nSample a batch of $(x,d)$ from training set;\\\\\nEstimate $\\hat{\\boldsymbol{\\theta}}_t$ from $P$ and input data $x$:\n$$\\hat{\\boldsymbol{\\theta}_t} = P(x)$$\\\\\nGenerate distribution $f_t(\\hat{\\boldsymbol{\\theta}}_t)$;\\\\\n$\\hat{d}_t = g^{\\min} $;\\\\\n$gap = g^{\\min} - g^{\\max}  $;\\\\ \n\\While {$gap > (g^{\\min}-g^{\\max})\\cdot10^{-6}$ }\n{\n$\\hat{d}_t = \\hat{d}_t - sign(C'(\\hat{d}_t)+\\tilde{R}'(\\hat{d}_t))\\cdot{gap}$;\\\\\n$gap = gap\/2$;\n}\n$\\hat{g}_t = \\text{med}\\left\\{g^{\\min}, \\hat{d}_t, g^{\\max}\\right\\}$;\\\\\nCompute generation policy $\\hat{\\boldsymbol{g}}_t$ from ${G}(\\hat{g}_t)$;\\\\\nCompute task loss $L_t(\\hat{\\boldsymbol{g}}_t,d)$\\\\\nCompute derivatives $\\frac{\\partial{L_t(\\hat{\\boldsymbol{g}}_t,d)}}{\\partial\\hat{{\\boldsymbol{\\theta}}}_t}$;\\\\\nBack propagate and update the predictor $P$;\\\\\nCompute the loss in validation set $L_v$ by $P$;\n}\n\\caption{Optimization Kernel Based  Learning}\n\\label{kernel} \n\\end{algorithm}\n\nWe want to close this section by making remarks on generalizing our framework to the ED problem with ramping constraints. The only difference is that the dispatch policies cannot be decoupled across time. Hence, the risk cost has to rely on $T$ variables, i.e., $g_1, ..., g_T$, instead of a single variable. In this case, we need to construct a high dimensional optimization kernel to enable the efficient learning.\n\n\\section{End-to-End Learning: Model-Free}\n\\label{modelfree}\nWhile the optimization kernel can effectively speed up the learning process, in this section, we highlight that the knowledge of distribution is great, but not essential.\n\nDenote $f(d_t;\\hat{\\boldsymbol{\\theta}}_t)$ and $F(d_t;\\hat{\\boldsymbol{\\theta}}_t)$ the \\emph{pdf} and \\emph{cdf} of hypothetical distribution of $\\hat{d}_t$, and denote $H(d_t)$ its true \\emph{cdf}.\n\nThus, the derivative of task loss ${L}_t(g_t)$ with respect to $g_t$ can be obtained as follows:\n\\begin{align}\n    \\frac{\\partial{{L}_t(g_t)}}{\\partial{g_t}} = (\\gamma_1 + \\gamma_2)F(g_t;\\hat{\\boldsymbol{\\theta}}_t) - \\gamma_1 + C'(g_t).\n\\end{align}\nCombining the first order optimality condition with the generation capacity constraint $g_t \\in [g_t^{\\min}, g_t^{\\max}]$, the optimal policy $g^0_t$ given the distribution $F(x;\\hat{\\boldsymbol{\\theta}}_t)$ can be obtained as follows.\n\\begin{align}\n    g_t^0 &= \\text{med}\\left\\{g_t^{\\min},F^{-1}\\left(\\frac{\\gamma_1-C'(g_t^0)}{\\gamma_1+\\gamma_2};\\hat{\\boldsymbol{\\theta}}_t\\right), g_t^{\\max}\\right\\}.\n    \\label{g0}\n\\end{align}\nNote that $\\text{med}$ is median operator. We denote $g_t^0 = g_t(\\hat{\\boldsymbol{\\theta}}_t)$ to highlight that it could be a function of the distribution parameters $\\hat{\\boldsymbol{\\theta}}_t$. The same analysis applies to decide the true optimal control policy $g_t^*$ (given $H(d_t)$):\n\\begin{align}\n    g_t^* &= \\text{med}\\left\\{g_t^{\\min},H^{-1}\\left(\\frac{\\gamma_1-C'(g_t^*)}{\\gamma_1+\\gamma_2}\\right), g_t^{\\max}\\right\\}.\n    \\label{g*}\n\\end{align}\nCombining Eqs. (\\ref{g0}) and (\\ref{g*}), we can examine the performance loss $\\Delta_L(g_t^*, g_t(\\hat{\\boldsymbol{\\theta}}_t))$ induced by the inaccurate distribution estimation:\n\\begin{equation}\n    \\begin{split}\n        \\Delta_L &= |{L}_t(\\hat{\\boldsymbol{\\theta}}_t)-{L}_t(g_t^*)|= \\left|\\int_{g_t^*}^{g_t(\\hat{\\boldsymbol{\\theta}}_t)}{L_t'(x)dx}\\right|\\\\\n    &= \\left|\\int_{g_t^*}^{g_t(\\hat{\\boldsymbol{\\theta}}_t)}{(\\gamma_1+\\gamma_2)H(x)-\\gamma_1+C_t'(x)dx}\\right|.\n    \\end{split}\n\\end{equation}\nDefine \n\\begin{align}\n   K(x) = (\\gamma_1 + \\gamma_2)H(x) - \\gamma_1 + C_t'(x),\n\\end{align}\nit is straightforward to verify the following lemma:\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Lemma 4}:\nThe function $K(x)$ is either always positive over $[g_t^* , g_t(\\hat{\\boldsymbol{\\theta}}_t)]$ or always negative over $[g_t(\\hat{\\boldsymbol{\\theta}}_t), g_t^*]$.\n\\vspace{0.1cm}\n\nWith the lemma, we can conclude that the ultimate goal of the learning process should be to estimate $g^*_t$ as accurately as possible, which does not require the explicit knowledge of distribution. This is also desirable according to Occam's Razor \\cite{Occam}: estimating the distribution often leads to more parameter estimations.\nSpecifically, when the data are not sufficiently large, the trained model may suffer from overfitting issues or poor generalization ability in practice.\n\nTo achieve an accurate prediction for $g^*_t$, we design the following loss function for the learning process:\n\\begin{align}\n    Q_t(\\hat{g}_t, d_t) \\!=\\! C_t(\\hat{g}_t)\\!-\\! C_t(d_t)\\!+\\!\\gamma_1(d_t\\!-\\! \\hat{g}_t)^+ \\!+\\! \\gamma_2(\\hat{g}_t- d_t)^+.\n    \\label{qt}\n\\end{align}\nWe want to emphasize that in Eq.(\\ref{qt}), $d_t$ is the actual demand during learning.\nAlso, the geographical distributed demand information has been encoded into the function $C_t(\\cdot)$ via $\\mathbb{E}[d_{jt}]$, $\\forall{j}$, together with all the transmission line capacity constraints. Based on this loss function, we design the model-free end-to-end learning framework as described in Algorithm \\ref{mdfree}.\n\\begin{algorithm}\n\\SetAlgoLined\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\n\\Input{training set and validation set $(x,d)$;}\n\\Output{predictor $P$;}\nCompute $C_t(g_t)$'s function curve;\n\nCompute $g^{\\min} $ and $g^{\\max} $;\n\n\\While{Loss in validation set $L_v$ doesn't increase}\n{\nSample a batch of $(x,d)$ from training set;\\\\\nCompute the total generation $\\hat{g_t}$ from predictor $P$ and input data $x$:\n$$\\hat{g_t} = P(x)$$\\\\\n$\\hat{g_t} = \\text{med}\\left\\{g^{\\min}, \\hat{g_t}, g^{\\max}\\right\\}$;\\\\\nCompute task loss $Q_t(\\hat{g_t},d)$;\\\\\nCompute derivatives $\\frac{\\partial{Q_t(\\hat{{g}}_t,d)}}{\\partial{\\hat{{g}}_t}}$;\\\\\nBack propagate and update the predictor $P$;\\\\\nCompute the loss in validation set $L_v$ by $P$\\;\n}\n\\caption{Optimization-Free Learning}\n\\label{mdfree} \n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical Studies}\n\\label{experiment}\nWe compare the performance of different approaches on a four-bus system in terms of effectiveness, robustness and efficiency. We further illustrate the effectiveness of the model-free framework on the IEEE 39-bus system.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=4cm]{bus.png}}\n\\caption{4-Bus System.}\n\\label{4bus}\n\\end{figure}\n\\begin{table}[t]\n\t\\centering\n\t\\caption{4-Bus System Parameters}\n\t\\begin{tabular}{ccccc}\n\t\t\\toprule \n\t\tBus-Bus  & 1 & 2  &  3 &  4 \\\\ \n\t\t\\midrule \n\t\tImpedance (p.u.) & $-j$ & $-j$ & $-j$ & $-j$ \\\\\n\t\t\\midrule \n\t\tCapacity (MW) & 1.5 & 1.5 & 1.5 & 1.5\\\\\n\t\t\\bottomrule \n\t\\end{tabular}\n\t\\label{4bus_grid}\n\\end{table}\n\\subsection{Setup for 4-Bus System}\nAs shown in Fig. \\ref{4bus}, the system contains 3 generators at Bus 1 to Bus 3, respectively, and a single load at Bus 4. To capture the stochastic nature of the net demand, we use the 5-year PJM load data from 2012 to 2016 \\cite{PJM}. We assume the marginal costs of the three generators are $\\$$40, $\\$$50, $\\$$60\/MWh, respectively.\nWe further set the unit shortage penalty $\\gamma_1$ to be $\\$$100\/MWh, and the unit excess penalty $\\gamma_2$ to be $\\$$10\/MWh. The impedances and the line capacities of the network can be found in Table \\ref{4bus_grid}.\n\nWe employ a 2-layer neural network with 128 neurons in each layer to implement our proposed frameworks. During the learning process, we divide the 5-year data set into 3 sets: training set, validation set, and test set. The training set contains the load data in the first $1,200$ days. The data in the following 200 days construct the validation set, and the data in the last 400 days form the test set.\n\nAll models are trained to the convergence of accuracy decided by the validation set.\nFor all the models, they share the same inputs: historical hourly load data in the former 24 hours and weekday-weekend type. \n\n\n\n\n\n\\subsection{Performance Evaluation: Effectiveness}\nWe compare the performance of four frameworks: conventional approach with MSE criteria, end-to-end learning with task specific criteria, end-to-end learning with optimization kernel, and model-free end-to-end framework. The evaluation metrics are load prediction error (MSE) and the loss in dispatch cost.\n\nDuring the comparison, we divide each day into four periods: midnight (from 0:00 am to 6:00 am), morning (from 6:00 am to 12:00 pm), afternoon (from 12:00 pm to 6:00 pm), and evening (from 6:00 pm to 0:00 am), and conduct the comparison for these four periods, respectively. Figure \\ref{difm} plots the comparison with respect to the two evaluation metrics.\n\nIt is interesting to observe that the conventional approach doesn't produce the best prediction in all periods. Yet it does incur relatively high loss in the dispatch cost (i.e., additional dispatch cost). As suggested by Fig.\\ref{difm}(b), all the other three frameworks improve the effectiveness of the conventional approach (benchmark). More precisely, task specific criteria achieves $1.53 \\%$ less loss than the benchmark; optimization kernel framework further improves the effectiveness by $2.13\\%$; and the model-free end-to-end learning framework achieves $1.27\\%$ additional improvement by relaxing the assumption of the load prediction distributions. This highlights the remarkable performance of our proposed frameworks.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=8cm]{error_finals120.jpg}}\n\\caption{Effectiveness Comparison (the shadowed areas illustrate the $\\pm0.1\\sigma$ zone for the four frameworks).}\n\\label{difm}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Performance Evaluation: Robustness}\nThe task specific framework and the optimization kernel framework both requires the knowledge on the type of prediction distribution for effective learning. We target to examine their robustness through synthetic data generated from Normal distribution, Uniform distribution, and Bounded Pareto distribution. We use the model-free framework as the benchmark.\nWe take the optimization kernel framework as an example. Fig. \\ref{robust} visualizes its robustness evaluation. It is suggested that the proposed framework is rather robust to most light tail distributions (e.g., Normal distribution and Uniform distribution) as the loss\/error in the two evaluation metrics is bounded by $4 \\%$ compared with the benchmark. The robustness is weakened facing heavy tail distributions (e.g., bounded Pareto distribution) yet the loss\/error is still bounded by $8 \\%$.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=8.5cm]{distribution_errors11.jpg}}\n\\caption{Robustness Evaluation for End-to-End Framework with Optimization Kernel.}\n\\label{robust}\n\\end{figure}\n\n\n\n\n\n\\subsection{Performance Evaluation: Efficiency}\nWhile the three proposed frameworks all outperform the conventional approach, the extra performance does not come free: the learning process is more complex and hence more time consuming than that of the conventional approach. Figure \\ref{time} illustrates the runing time for the four frameworks on a log-scale. Compared with the task specific framework, the optimization kernel speeds up the learning process by $182\\%$, which verifies the effectiveness of the kernel. The model-free framework further speeds up the process to the comparable level of the conventional approach!\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=5cm]{TCOS.jpg}}\n\\caption{Efficiency Comparison.}\n\\label{time}\n\\end{figure}\n\n\n\\subsection{Performance Evaluation for IEEE 39-bus System}\nWe further verify the performance of the model-free framework on a larger system: the IEEE 39-bus system. We follow the network parameters in the system as suggested in \\cite{39bus}. We set the marginal cost of the generators from bus 30 to bus 39 in an increasing order: from $\\$30$\/MWh up to $\\$48$\/MWh at a step size of $\\$2$\/MWh. As such, we set unit shortage penalty $\\gamma_1$ to be $\\$50$\/MWh and the unit excess penalty to be $\\$2$\/MWh.\n\nFigure \\ref{IEEE39} illustrates the performance improvement of the model-free framework, compared with the conventional approach as a benchmark. While our framework generates $72.85\\%$ more load prediction error, it does reduce the loss in dispatch cost by $8.24\\%$.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=8.5cm]{error_ieee39_finalsq.jpg}}\n\\caption{Performance Evaluation for IEEE 39-Bus System  (the shadowed areas illustrate the $\\pm0.1\\sigma$ zone).}\n\\label{IEEE39}\n\\end{figure}\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{conclude}\nIn this paper, we seek to design an effective end-to-end machine learning framework for ED. Specifically, motivated by the task specific criteria, we design the optimization kernel to speed up the learning process, which ultimately leads to our model-free framework. Numerical studies verify the efficiency and effectiveness of our proposed schemes.\n\nOur work can be extended in various interesting ways. For example, we consider the network constraints satisfies on expectation. That is, we implicitly assume the prediction error can be bounded within a relatively small range, which won't affect the network constraints too much. It will be interesting to propose a more adaptive way to ensure the network constraints in the end-to-end learning framework.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}