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@ -1,6 +1,6 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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<!--Converted with LaTeX2HTML 2008 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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* with significant contributions from:
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@ -13,7 +13,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<META NAME="resource-type" CONTENT="document">
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<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
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@ -28,20 +28,16 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@ -69,21 +65,21 @@ PDE.
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Each point of the discretization mesh will have (at least) one
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Each point of the discretization mesh will have (at least) one
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associated equation/variable, and therefore one index. We say that
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associated equation/variable, and therefore one index. We say that
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point <IMG
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point <IMG
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WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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SRC="img4.png"
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ALT="$i$"> <EM>depends</EM> on point <IMG
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ALT="$i$"> <EM>depends</EM> on point <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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SRC="img5.png"
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ALT="$j$"> if the equation for a
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ALT="$j$"> if the equation for a
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variable associated with <IMG
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variable associated with <IMG
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WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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SRC="img4.png"
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ALT="$i$"> contains a term in <IMG
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ALT="$i$"> contains a term in <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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SRC="img5.png"
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ALT="$j$">, or equivalently
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ALT="$j$">, or equivalently
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if <IMG
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if <IMG
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WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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SRC="img6.png"
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ALT="$a_{ij} \ne0$">.
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ALT="$a_{ij} \ne0$">.
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After the partition of the discretization mesh into <EM>sub-domains</EM>
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After the partition of the discretization mesh into <EM>sub-domains</EM>
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@ -136,19 +132,19 @@ work [<A
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<P>
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<P>
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We denote the sets of internal, boundary and halo points for a given
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We denote the sets of internal, boundary and halo points for a given
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subdomain by <IMG
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subdomain by <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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SRC="img7.png"
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ALT="$\cal I$">, <IMG
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ALT="$\cal I$">, <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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SRC="img8.png"
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ALT="$\cal B$"> and <IMG
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ALT="$\cal B$"> and <IMG
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WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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SRC="img9.png"
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ALT="$\cal H$">.
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ALT="$\cal H$">.
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Each subdomain is assigned to one process; each process usually
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Each subdomain is assigned to one process; each process usually
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owns one subdomain, although the user may choose to assign more than
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owns one subdomain, although the user may choose to assign more than
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one subdomain to a process. If each process <IMG
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one subdomain to a process. If each process <IMG
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WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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SRC="img4.png"
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ALT="$i$"> owns one
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ALT="$i$"> owns one
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subdomain, the number of rows in the local sparse matrix is
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subdomain, the number of rows in the local sparse matrix is
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@ -156,7 +152,7 @@ subdomain, the number of rows in the local sparse matrix is
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$|{\cal I}_i| + |{\cal B}_i|$
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$|{\cal I}_i| + |{\cal B}_i|$
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-->
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-->
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<IMG
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<IMG
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WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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SRC="img10.png"
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ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert$">, and the number of local columns
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ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert$">, and the number of local columns
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(i.e. those for which there exists at least one non-zero entry in the
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(i.e. those for which there exists at least one non-zero entry in the
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@ -164,7 +160,7 @@ local rows) is <!-- MATH
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$|{\cal I}_i| + |{\cal B}_i| +|{\cal H}_i|$
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$|{\cal I}_i| + |{\cal B}_i| +|{\cal H}_i|$
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-->
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-->
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<IMG
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<IMG
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WIDTH="117" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="118" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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SRC="img11.png"
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ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert +\vert{\cal H}_i\vert$">.
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ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert +\vert{\cal H}_i\vert$">.
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@ -203,20 +199,16 @@ points in the literature.
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Salvatore Filippone
15 years ago
