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Fixup the documentation.

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Salvatore Filippone
2018-05-14 14:32:48 +01:00
parent 8ee76a1a82
commit 4b43164668
138 changed files with 1396 additions and 1295 deletions
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docs/html/node15.html
+17 -15
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@@ -1,6 +1,6 @@
<!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>
<HEAD>
<TITLE>Smoothers and coarsest-level solvers</TITLE>
@@ -9,7 +9,7 @@
<META NAME="resource-type" CONTENT="document">
<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">
<LINK REL="STYLESHEET" HREF="userhtml.css">
@@ -55,8 +55,8 @@ Smoothers and coarsest-level solvers
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
a hybrid version of the forward and backward Gauss-Seidel methods, and the
additive Schwarz (AS) ones (see, e.g., [<A
HREF="node36.html#Saad_book">20</A>,<A
HREF="node36.html#dd2_96">21</A>]).
HREF="node36.html#Saad_book">21</A>,<A
HREF="node36.html#dd2_96">22</A>]).
</BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The hybrid Gauss-Seidel
@@ -68,7 +68,7 @@ the beginning of the current iteration.
</BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In the AS methods, the index space <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"></SPAN> is divided into <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
@@ -78,11 +78,11 @@ subsets <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$"></SPAN> of size <SPAN CLASS="MATH"><IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$n_{k,i}$"></SPAN>, possibly
overlapping. For each <SPAN CLASS="MATH"><IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$"></SPAN> we consider the restriction
operator <!-- MATH
@@ -93,13 +93,13 @@ operator <!-- MATH
SRC="img53.png"
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$"></SPAN>
that maps a vector <SPAN CLASS="MATH"><IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$"></SPAN> to the vector <SPAN CLASS="MATH"><IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$x_i^k$"></SPAN> made of the components of <SPAN CLASS="MATH"><IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$"></SPAN>
with indices in <SPAN CLASS="MATH"><IMG
@@ -120,7 +120,7 @@ with indices in <SPAN CLASS="MATH"><IMG
WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$A_i^k=R_i^kA^kP_i^k$"></SPAN>, which is the restriction of <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"></SPAN> to the index
space <SPAN CLASS="MATH"><IMG
@@ -141,9 +141,11 @@ The classical AS preconditioner <SPAN CLASS="MATH"><IMG
-->
<IMG
WIDTH="219" HEIGHT="59" BORDER="0"
WIDTH="218" HEIGHT="59" BORDER="0"
SRC="img59.png"
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
ALT="\begin{displaymath}
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
@@ -170,7 +172,7 @@ involves
SRC="img62.png"
ALT="$\Omega_i^k$"></SPAN> and of the corresponding
operators <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$R_i^k$"></SPAN> (and <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
@@ -203,13 +205,13 @@ multilevel application phase, requires
</BIG></BIG></BIG>
<UL>
<LI>the restriction of <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
SRC="img67.png"
ALT="$w^k$"></SPAN> to the subspaces <!-- MATH
$\mathbb{R}^{n_{k,i}}$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="$\mathbb{R}^{n_{k,i}}$"></SPAN>,
i.e. <!-- MATH