|
|
|
@ -1,6 +1,6 @@
|
|
|
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
|
|
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
|
|
|
|
|
|
|
|
|
|
|
<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
|
|
|
|
<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
|
|
|
|
<HTML>
|
|
|
|
<HTML>
|
|
|
|
<HEAD>
|
|
|
|
<HEAD>
|
|
|
|
<TITLE>Smoothed Aggregation</TITLE>
|
|
|
|
<TITLE>Smoothed Aggregation</TITLE>
|
|
|
|
@ -9,7 +9,7 @@
|
|
|
|
<META NAME="resource-type" CONTENT="document">
|
|
|
|
<META NAME="resource-type" CONTENT="document">
|
|
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
|
|
|
|
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
|
|
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
|
|
|
|
|
|
|
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
|
|
|
@ -54,27 +54,27 @@ Smoothed Aggregation
|
|
|
|
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
|
|
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
|
|
<P>
|
|
|
|
<P>
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img25.png"
|
|
|
|
SRC="img25.png"
|
|
|
|
ALT="$P^k$"></SPAN>, used to compute
|
|
|
|
ALT="$P^k$"></SPAN>, used to compute
|
|
|
|
the coarse-level matrix <SPAN CLASS="MATH"><IMG
|
|
|
|
the coarse-level matrix <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img15.png"
|
|
|
|
SRC="img15.png"
|
|
|
|
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
|
|
|
|
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
|
|
|
|
algorithm described in [<A
|
|
|
|
algorithm described in [<A
|
|
|
|
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
|
|
|
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
|
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>].
|
|
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>].
|
|
|
|
The basic idea of this algorithm is to build a coarse set of indices
|
|
|
|
The basic idea of this algorithm is to build a coarse set of indices
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img26.png"
|
|
|
|
SRC="img26.png"
|
|
|
|
ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img9.png"
|
|
|
|
SRC="img9.png"
|
|
|
|
ALT="$\Omega^k$"></SPAN> into disjoint
|
|
|
|
ALT="$\Omega^k$"></SPAN> into disjoint
|
|
|
|
subsets (aggregates), and to define the coarse-to-fine space transfer operator
|
|
|
|
subsets (aggregates), and to define the coarse-to-fine space transfer operator
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img25.png"
|
|
|
|
SRC="img25.png"
|
|
|
|
ALT="$P^k$"></SPAN> by applying a suitable smoother to a simple piecewise constant
|
|
|
|
ALT="$P^k$"></SPAN> by applying a suitable smoother to a simple piecewise constant
|
|
|
|
prolongation operator, with the aim of improving the quality of the coarse-space correction.
|
|
|
|
prolongation operator, with the aim of improving the quality of the coarse-space correction.
|
|
|
|
@ -84,26 +84,26 @@ prolongation operator, with the aim of improving the quality of the coarse-space
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
<OL>
|
|
|
|
<OL>
|
|
|
|
<LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
|
|
|
|
<LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img9.png"
|
|
|
|
SRC="img9.png"
|
|
|
|
ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img26.png"
|
|
|
|
SRC="img26.png"
|
|
|
|
ALT="$\Omega^{k+1}$"></SPAN>;
|
|
|
|
ALT="$\Omega^{k+1}$"></SPAN>;
|
|
|
|
</LI>
|
|
|
|
</LI>
|
|
|
|
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
|
|
|
|
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img25.png"
|
|
|
|
SRC="img25.png"
|
|
|
|
ALT="$P^k$"></SPAN>;
|
|
|
|
ALT="$P^k$"></SPAN>;
|
|
|
|
</LI>
|
|
|
|
</LI>
|
|
|
|
<LI>application of <SPAN CLASS="MATH"><IMG
|
|
|
|
<LI>application of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img25.png"
|
|
|
|
SRC="img25.png"
|
|
|
|
ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img17.png"
|
|
|
|
SRC="img17.png"
|
|
|
|
ALT="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img15.png"
|
|
|
|
SRC="img15.png"
|
|
|
|
ALT="$A^{k+1}$"></SPAN>.
|
|
|
|
ALT="$A^{k+1}$"></SPAN>.
|
|
|
|
</LI>
|
|
|
|
</LI>
|
|
|
|
@ -111,18 +111,18 @@ prolongation operator, with the aim of improving the quality of the coarse-space
|
|
|
|
<P>
|
|
|
|
<P>
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
|
|
|
|
described in [<A
|
|
|
|
described in [<A
|
|
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
|
|
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] is used. In this algorithm,
|
|
|
|
each index <!-- MATH
|
|
|
|
each index <!-- MATH
|
|
|
|
$j \in \Omega^{k+1}$
|
|
|
|
$j \in \Omega^{k+1}$
|
|
|
|
-->
|
|
|
|
-->
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img27.png"
|
|
|
|
SRC="img27.png"
|
|
|
|
ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img28.png"
|
|
|
|
SRC="img28.png"
|
|
|
|
ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img9.png"
|
|
|
|
SRC="img9.png"
|
|
|
|
ALT="$\Omega^k$"></SPAN>,
|
|
|
|
ALT="$\Omega^k$"></SPAN>,
|
|
|
|
consisting of a suitably chosen index <!-- MATH
|
|
|
|
consisting of a suitably chosen index <!-- MATH
|
|
|
|
@ -133,7 +133,7 @@ consisting of a suitably chosen index <!-- MATH
|
|
|
|
SRC="img29.png"
|
|
|
|
SRC="img29.png"
|
|
|
|
ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
|
|
|
|
ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
|
|
|
|
strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
|
|
|
|
strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img30.png"
|
|
|
|
SRC="img30.png"
|
|
|
|
ALT="$i$"></SPAN>, i.e.,
|
|
|
|
ALT="$i$"></SPAN>, i.e.,
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
@ -149,11 +149,13 @@ strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
|
|
|
|
<A NAME="eq:strongly_coup"></A>
|
|
|
|
<A NAME="eq:strongly_coup"></A>
|
|
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
|
|
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
|
|
|
|
WIDTH="387" HEIGHT="72" BORDER="0"
|
|
|
|
WIDTH="387" HEIGHT="49" BORDER="0"
|
|
|
|
SRC="img31.png"
|
|
|
|
SRC="img31.png"
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r ...
|
|
|
|
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
|
|
|
|
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
\end{displaymath}"></TD>
|
|
|
|
\left\{ r \i...
|
|
|
|
|
|
|
|
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
|
|
|
|
|
|
|
|
\end{displaymath}"></TD>
|
|
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
|
|
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
|
|
|
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
|
|
|
</TABLE>
|
|
|
|
</TABLE>
|
|
|
|
@ -162,10 +164,10 @@ for a given threshold <!-- MATH
|
|
|
|
$\theta \in [0,1]$
|
|
|
|
$\theta \in [0,1]$
|
|
|
|
-->
|
|
|
|
-->
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img32.png"
|
|
|
|
SRC="img32.png"
|
|
|
|
ALT="$\theta \in [0,1]$"></SPAN> (see [<A
|
|
|
|
ALT="$\theta \in [0,1]$"></SPAN> (see [<A
|
|
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
|
|
|
|
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] for the details).
|
|
|
|
Since this algorithm has a sequential nature, a decoupled
|
|
|
|
Since this algorithm has a sequential nature, a decoupled
|
|
|
|
version of it is applied, where each processor independently executes
|
|
|
|
version of it is applied, where each processor independently executes
|
|
|
|
the algorithm on the set of indices assigned to it in the initial data
|
|
|
|
the algorithm on the set of indices assigned to it in the initial data
|
|
|
|
@ -180,11 +182,11 @@ MLD2P4, since it has been shown to produce good results in practice
|
|
|
|
[<A
|
|
|
|
[<A
|
|
|
|
HREF="node36.html#aaecc_07">5</A>,<A
|
|
|
|
HREF="node36.html#aaecc_07">5</A>,<A
|
|
|
|
HREF="node36.html#apnum_07">7</A>,<A
|
|
|
|
HREF="node36.html#apnum_07">7</A>,<A
|
|
|
|
HREF="node36.html#TUMINARO_TONG">24</A>].
|
|
|
|
HREF="node36.html#TUMINARO_TONG">25</A>].
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
<P>
|
|
|
|
<P>
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img25.png"
|
|
|
|
SRC="img25.png"
|
|
|
|
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
|
|
|
|
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
|
|
|
|
<!-- MATH
|
|
|
|
<!-- MATH
|
|
|
|
@ -210,10 +212,14 @@ MLD2P4, since it has been shown to produce good results in practice
|
|
|
|
<A NAME="eq:tent_prol"></A>
|
|
|
|
<A NAME="eq:tent_prol"></A>
|
|
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
|
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
|
|
|
WIDTH="287" HEIGHT="51" BORDER="0"
|
|
|
|
WIDTH="286" HEIGHT="52" BORDER="0"
|
|
|
|
SRC="img34.png"
|
|
|
|
SRC="img34.png"
|
|
|
|
ALT="\begin{displaymath}
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
\left\{...
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
...ega^k_j,
0 & \quad \mbox{otherwise},
\end{array} \right.
|
|
|
|
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
|
|
|
|
|
|
|
|
\left\{...
|
|
|
|
|
|
|
|
...Omega^k_j, \\
|
|
|
|
|
|
|
|
0 & \quad \mbox{otherwise},
|
|
|
|
|
|
|
|
\end{array} \right.
|
|
|
|
\end{displaymath}"></TD>
|
|
|
|
\end{displaymath}"></TD>
|
|
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
|
|
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
|
|
|
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
|
|
|
@ -223,21 +229,21 @@ where <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img28.png"
|
|
|
|
SRC="img28.png"
|
|
|
|
ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img9.png"
|
|
|
|
SRC="img9.png"
|
|
|
|
ALT="$\Omega^k$"></SPAN>
|
|
|
|
ALT="$\Omega^k$"></SPAN>
|
|
|
|
corresponding to the index <!-- MATH
|
|
|
|
corresponding to the index <!-- MATH
|
|
|
|
$j \in \Omega^{k+1}$
|
|
|
|
$j \in \Omega^{k+1}$
|
|
|
|
-->
|
|
|
|
-->
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img27.png"
|
|
|
|
SRC="img27.png"
|
|
|
|
ALT="$j \in \Omega^{k+1}$"></SPAN>.
|
|
|
|
ALT="$j \in \Omega^{k+1}$"></SPAN>.
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img25.png"
|
|
|
|
SRC="img25.png"
|
|
|
|
ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img35.png"
|
|
|
|
SRC="img35.png"
|
|
|
|
ALT="$\bar{P}^k$"></SPAN> a smoother
|
|
|
|
ALT="$\bar{P}^k$"></SPAN> a smoother
|
|
|
|
<!-- MATH
|
|
|
|
<!-- MATH
|
|
|
|
@ -257,9 +263,11 @@ P^k = S^k \bar{P}^k,
|
|
|
|
-->
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
<IMG
|
|
|
|
WIDTH="90" HEIGHT="30" BORDER="0"
|
|
|
|
WIDTH="91" HEIGHT="30" BORDER="0"
|
|
|
|
SRC="img37.png"
|
|
|
|
SRC="img37.png"
|
|
|
|
ALT="\begin{displaymath}
P^k = S^k \bar{P}^k,
\end{displaymath}">
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
|
|
|
P^k = S^k \bar{P}^k,
|
|
|
|
|
|
|
|
\end{displaymath}">
|
|
|
|
</DIV>
|
|
|
|
</DIV>
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
@ -267,9 +275,9 @@ in order to remove nonsmooth components from the range of the prolongator,
|
|
|
|
and hence to improve the convergence properties of the multilevel
|
|
|
|
and hence to improve the convergence properties of the multilevel
|
|
|
|
method [<A
|
|
|
|
method [<A
|
|
|
|
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
|
|
|
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
|
|
|
HREF="node36.html#Stuben_01">23</A>].
|
|
|
|
HREF="node36.html#Stuben_01">24</A>].
|
|
|
|
A simple choice for <SPAN CLASS="MATH"><IMG
|
|
|
|
A simple choice for <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img38.png"
|
|
|
|
SRC="img38.png"
|
|
|
|
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
|
|
|
|
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
@ -282,24 +290,26 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
|
|
|
-->
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
<IMG
|
|
|
|
WIDTH="175" HEIGHT="31" BORDER="0"
|
|
|
|
WIDTH="176" HEIGHT="32" BORDER="0"
|
|
|
|
SRC="img39.png"
|
|
|
|
SRC="img39.png"
|
|
|
|
ALT="\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
\end{displaymath}">
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
|
|
|
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
|
|
|
|
|
|
|
\end{displaymath}">
|
|
|
|
</DIV>
|
|
|
|
</DIV>
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
where <SPAN CLASS="MATH"><IMG
|
|
|
|
where <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img40.png"
|
|
|
|
SRC="img40.png"
|
|
|
|
ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img41.png"
|
|
|
|
SRC="img41.png"
|
|
|
|
ALT="$A^k$"></SPAN>,
|
|
|
|
ALT="$A^k$"></SPAN>,
|
|
|
|
<!-- MATH
|
|
|
|
<!-- MATH
|
|
|
|
$A^k_F = (\bar{a}_{ij}^k)$
|
|
|
|
$A^k_F = (\bar{a}_{ij}^k)$
|
|
|
|
-->
|
|
|
|
-->
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img42.png"
|
|
|
|
SRC="img42.png"
|
|
|
|
ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
|
|
|
|
ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
@ -321,17 +331,20 @@ where <SPAN CLASS="MATH"><IMG
|
|
|
|
<A NAME="eq:filtered"></A>
|
|
|
|
<A NAME="eq:filtered"></A>
|
|
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
|
|
|
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
|
|
|
WIDTH="514" HEIGHT="74" BORDER="0"
|
|
|
|
WIDTH="499" HEIGHT="59" BORDER="0"
|
|
|
|
SRC="img43.png"
|
|
|
|
SRC="img43.png"
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
\bar{a}_{ij}^k =
\left \{ \begin{array}{ll}
a_{ij}^k & ...
|
|
|
|
\bar{a}_{ij}^k =
|
|
|
|
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
\end{displaymath}"></TD>
|
|
|
|
\left \{ \begin{array}{ll}
|
|
|
|
|
|
|
|
a_{ij}^k & \m...
|
|
|
|
|
|
|
|
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
|
|
|
|
|
|
|
|
\end{displaymath}"></TD>
|
|
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
|
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
|
|
|
(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
|
|
|
|
(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
|
|
|
|
</TABLE>
|
|
|
|
</TABLE>
|
|
|
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
and <SPAN CLASS="MATH"><IMG
|
|
|
|
and <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img44.png"
|
|
|
|
SRC="img44.png"
|
|
|
|
ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
@ -360,14 +373,14 @@ of <SPAN CLASS="MATH"><IMG
|
|
|
|
SRC="img46.png"
|
|
|
|
SRC="img46.png"
|
|
|
|
ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
|
|
|
|
ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
|
|
|
|
problems, filtering the matrix <SPAN CLASS="MATH"><IMG
|
|
|
|
problems, filtering the matrix <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img41.png"
|
|
|
|
SRC="img41.png"
|
|
|
|
ALT="$A^k$"></SPAN> has little or no effect, and
|
|
|
|
ALT="$A^k$"></SPAN> has little or no effect, and
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
SRC="img41.png"
|
|
|
|
SRC="img41.png"
|
|
|
|
ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
|
|
|
|
ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
|
|
|
|
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
SRC="img49.png"
|
|
|
|
SRC="img49.png"
|
|
|
|
ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
|
|
|
|
ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
Salvatore Filippone
7 years ago
