\batchmode
























%














%


















%





%


\newcounter{c1}

\newcounter{c2}

%










\newfont{\sqi}{cmssqi8}










%
































\documentstyle[a4wide,psfig,11pt,html]{article}
%


\def\divg{\mbox{div}\,}

\def\grad{\mbox{grad}\,}

\def\rot{\mbox{rot}\,}

\def\curl{\mbox{curl}\,}

\def\tr{\mbox{tr}\,}

\def\deriv#1#2{\frac{d #1}{d #2}}

\def\lderiv#1#2{d #1 / d #2}

\def\pderiv#1#2{\frac{\partial #1}{\partial #2}}

\def\plderiv#1#2{\partial #1 / \partial #2}

\def\pderivxy#1#2#3{\frac{\partial #1}{\partial #2 \partial #3}}

\def\plderivxy#1#2#3{\partial #1 / \partial #2 \partial #3}

\def\derivmat#1#2{\frac{D#1}{D#2}}

\def\lderivmat#1#2{D#1/D#2}

\def\derivdel#1#2{\frac{\delta#1}{\delta#2}}

\def\lderivdel#1#2{\delta#1/\delta#2}

\def\derivcal#1#2{\frac{{\cal D}#1}{{\cal D}#2}}

\def\lderivcal#1#2{{\cal D}#1/{\cal D}#2}

\def\stackboxx#1{\stackrel{\scriptscriptstyle\Box}{#1}}

\def\stackboxxi#1{\stackrel{\scriptscriptstyle\mbox{\hspace*{0.2em}}
    \Box}{#1}}

\def\stackboxxii#1{\stackrel{\scriptscriptstyle
    \mbox{\hspace*{0.15em}}\Box}{#1}}

\def\stacktridown#1{\stackrel{\scriptscriptstyle
            \rule[-0.4ex]{0ex}{1ex}\bigtriangledown}{#1}}

\def\stacktriup#1{\stackrel{\scriptscriptstyle\bigtriangleup}{#1}}

\def\stackcirc#1{\stackrel{\circ}{#1}}

\def\stackstar#1{\stackrel{\star}{#1}}

\def\niet{\mbox{\hspace*{-0.2ex}}\!\!\!\!\!/\,\,\hspace*{0.2ex}}

\def\nga{=\niet}

\def\bm#1{\mbox{\boldmath $#1$}}

\def\pmb#1{\setbox0=\hbox{$#1$}
  \kern-.025em\copy0\kern-\wd0
  \kern .05 em\copy0\kern-\wd0
  \kern-.025em\raise.0433em\box0 }

\def\vm#1{\pmb{\bm{#1}}}

\newenvironment {enumroman}{\begin{list}{\roman{c1}.}{\usecounter{c1}}}%
{\end{list}}

\newenvironment {enumalpha}{\begin{list}{\alph{c2}.}{\usecounter{c2}}}%
{\end{list}}

\def\IR{\mbox{$I\!\!R$}}

\def\IP{\mbox{$I\!\!P$}}

\def\IN{\mbox{$I\!\!N$}}

\def\ICxi{\mbox{$C\!\!\!\!\raisebox{.75pt}{\mbox{\sqi{I}}}\,\,\,$}}

\def\II{\mbox{$I\!\!I$}}

\def\IE{\mbox{$I\!\!E$}}

\def\IEscr{\mbox{$\scriptstyle I\!\!E$}}

\def\IZ{\mbox{$Z\!\!\!Z$}}

\def\IZscr{\mbox{$\scriptstyle Z\!\!\!Z$}}

\def\chris#1#2#3{\{\!\!\!\begin{array}{c}
      \begin{array}{c}#1 \end{array}\\ 
      #2 #3 \end{array}\!\!\!\}}

\def\beq{\begin{equation}}

\def\eeq{\end{equation}}

\def\bq{\begin{quote}}

\def\eq{\end{quote}}

\def\bqt{\begin{quotation}}

\def\eqt{\end{quotation}}

\def\bea{\begin{eqnarray}}

\def\eea{\end{eqnarray}}

\def\beas{\begin{eqnarray*}}

\def\eeas{\end{eqnarray*}}

\def\ba{\begin{array}}

\def\ea{\end{array}}

\def\bit{\begin{itemize}}

\def\eit{\end{itemize}}

\def\bd{\begin{description}}

\def\ed{\end{description}}

\def\ben{\begin{enumerate}}

\def\een{\end{enumerate}}

\def\benr{\begin{enumroman}}

\def\eenr{\end{enumroman}}

\def\bena{\begin{enumalpha}}

\def\eena{\end{enumalpha}}

\def\bc{\begin{center}}

\def\ec{\end{center}}

\def\bfig{\begin{figure}}

\def\efig{\end{figure}}

\def\btab{\begin{tabbing}}

\def\etab{\end{tabbing}}

\def\btbl{\begin{tabular}}

\def\etbl{\end{tabular}}

\newcommand {\censored}[1]{}

\newcommand {\styleart}[1]{{\it #1}}

\newcommand {\stylebk}[1]{{\it #1}}

\newcommand {\article}[5]
{
  %
  %
  %
  %
  %


  \bibitem{#1}   
  %
  \begin{quote}
    #2,          
    `#3',
    \styleart{#4},
    #5.              
  \end{quote}
}

\newcommand {\book}[5]
{
  \bibitem{#1}     %
  \begin{quote}
    #2,            %
    \stylebk{#3},  %
    \stylebk{#4}   %
    #5.            %
  \end{quote}
}

\newenvironment {xequation}{ 
  \begin{equation}}{
  \end{equation} 
}

\newenvironment {xarray}[1]{  
  \begin{array}{#1}}{ 
  \end{array} 
}

\newenvironment {accolade}[1]{ 
  \begin{xequation}
  \left\{
  \begin{xarray}{#1}}{
  \end{xarray}
  \right.
  \end{xequation} 
}

\newenvironment {xmatrix}[1]{ 
  \left[
    \begin{xarray}{#1}}{
    \end{xarray}
  \right] 
}

\newcommand {\xmatrixone}[1]{ 
  {\left[
    \begin{array}{c}
      #1
    \end{array}
  \right] } 
}

\newcommand {\xmatrixtwo}[1]{ 
  {\left[
    \begin{array}{cc}
      #1
    \end{array}
  \right] } 
}

\newcommand {\xmatrixthree}[1]{ 
  {\left[
    \begin{array}{ccc}
      #1
    \end{array}
  \right] } 
}

\newcommand {\xmatrixfour}[1]{ 
  {\left[
    \begin{array}{cccc}
      #1
    \end{array}
  \right] } 
}

\newcommand {\xmatrixfive}[1]{ 
  {\left[
    \begin{array}{ccccc}
      #1
    \end{array}
  \right] } 
}

\newenvironment {accoladedisplay}[1]{ 
  \begin{displaymath}
  \left\{
    \begin{xarray}{#1}}{
    \end{xarray}
  \right.
  \end{displaymath} 
}

\newenvironment {sumup}[2]{ 
  \begin{xequation}
  #1
  \left\{
    \begin{xarray}{#2} }{
    \end{xarray}
  \right.
  \end{xequation} 
}

\newenvironment {proof}{ 
  \theoremstyle
  {\bf Proof:}
  \samepage
  \begin{quote}}{
  \end{quote}
  \flushleft $\Box$ \\ [2ex] 
}

\newcommand {\USEunderscore}{\catcode`\_=13}

\newcommand {\usc}{{\begin{USEunderscore}{_}\end{USEunderscore}}}

\newcommand {\tablestyle}{\sf \small}

\newcommand {\theoremstyle}{\sl}

\newcommand {\real}{\mbox{\sl I$\!$R$\!\!$ }}

\newcommand {\complex}{\mbox{\sl I$\!$C$\!\!$ }}

\newcommand {\nat}{\mbox{\sl I$\!$N$\!\!$ }}

\newcommand {\Reyn}{\mbox{Re}}

\newcommand {\isnas}{ISNaS}

\newcommand {\reduce}{REDUCE}

\newcommand {\liss}{ 
  \mbox{
    $\mbox{L}_i\mbox{SS}$
  } 
}

\newcommand {\half}{\frac{1}{2}}

\newcommand {\xref}[1]{(\ref{#1})}

\newcommand {\figref}[1]{\ref{#1}}

\newcommand {\dualgraph}{{\mbox{$\cal G$}}}

\newcommand {\neighbor}[1]{ {\mbox{${\cal N}_{#1}$}}
}

\newcommand {\boundary}[1]{ {\mbox{${\cal B}_{#1}$}}}

\newcommand {\internal}[1]{
  \stackrel{\circ}{#1}
}

\newcommand {\Bomega}{{\partial \Omega}}

\newcommand {\restr}[1]{ R_{#1} }

\newcommand {\prol}[1]{ P_{#1} }

\newcommand {\rstok}[1]{ {\left. {#1} \right|} }

\newcommand {\vanish}[1]{}

\renewcommand {\pderiv}[2]{
      \frac{\displaystyle \partial^{\,\!} #1}{\displaystyle \partial_{\,\!} #2}
}

\renewcommand {\deriv}[2]{
      \frac{\displaystyle \mbox{d}^{\,\!} #1}{\displaystyle \mbox{d}_{\,\!} #2}
}

\newcommand {\derivn}[3]{
      \frac{\displaystyle \mbox{d}^{#1} #2}{\displaystyle \mbox{d}_{\,\!} #3^{#1}}
}

\newcommand {\lamda}{\lambda}

\newcommand {\lamdba}{\lambda}

\newcommand {\alfa}{\alpha}

\newcommand {\pe}{\mbox{Pe}}

\newcommand {\infnorm}[1]{ \| #1 \|_{\infty} }

\newcommand {\eps}{\epsilon}

\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathaccent "707E\relax {N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1471: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$4\times 4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1473: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}-\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} x^2} 
  -\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} y^2} + a^1 \frac{\displaystyle \partial^{\,\!} u}{\displaystyle \partial_{\,\!} x} + a^2\frac{\displaystyle \partial^{\,\!} u}{\displaystyle \partial_{\,\!} y} 
  + c u = f \mbox{\ on \ } \Omega = [-1,1]\times [-1,1].  \label{eq-des-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a^1 = a^2 = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1475: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon = 10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1477: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}u^{m+1} = (I - N^{-1}A) u^m + N^{-1} f.    \label{eq-acc-2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1481: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}A =
  {\left[
\begin{array}{ccc}

    A_{11} & A_{12} & A_{13} \\  
    A_{21} & A_{22} & A_{23} \\  
    A_{31} & A_{32} & A_{33} 
  
\end{array}
\right] }.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1493: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}N = N_{gs} = {\left[
\begin{array}{ccc}

        A_{11} & \emptyset & \emptyset \\  
        A_{21} & A_{22}    & \emptyset \\  
        A_{31} & A_{32}    & A_{33}         
      
\end{array}
\right] } \mbox{ \ or \ }
  N = N_{jac} = {\left[
\begin{array}{ccc}

        A_{11} & \emptyset & \emptyset \\  
        \emptyset & A_{22}    & \emptyset \\  
        \emptyset & \emptyset & A_{33}       
      
\end{array}
\right] }                                        \label{eq-gs-jac}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_{gs}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1497: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_{jac}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1499: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1503: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}(I-N^{-1}A) u = (I-N^{-1}A) Q v      \label{eq-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q = {\left[
\begin{array}{c}
 0 \\   I 
\end{array}
\right] }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1515: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u = {\left[
\begin{array}{c}
0 \\   v
\end{array}
\right] }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1519: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(I-N^{-1}A) {\left[
\begin{array}{c}
w \\  0
\end{array}
\right] } = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1521: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u^{m+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1523: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{
  \psfig{file=relvars.eps,width=7.8cm,height=5.0cm}
}

\label{fig-relvars}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1531: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}v^{m+1} = Q^T u^{m+1} = Q^T(I-N^{-1}A) Q v^m  + Q^T N^{-1} f.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}v = Q^T(I-N^{-1}A) Q v  + Q^T N^{-1} f.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Q^T N^{-1}A Q v = Q^T N^{-1}f. \label{eq-interface}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1537: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\tilde{N} = \tilde{N}_{gs} = {\left[
\begin{array}{ccc}

        \tilde{A}_{11} & \emptyset & \emptyset \\  
        A_{21}         & \tilde{A}_{22}    & \emptyset \\  
        A_{31}         & A_{32}    & \tilde{A}_{33}         
      
\end{array}
\right] } \mbox{ \ or \ }
  \tilde{N} = \tilde{N}_{jac} = {\left[
\begin{array}{ccc}

        \tilde{A}_{11} & \emptyset & \emptyset \\  
        \emptyset & \tilde{A}_{22}    & \emptyset \\  
        \emptyset & \emptyset & \tilde{A}_{33}       
      
\end{array}
\right] }                     .                  
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}_{gs}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1541: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}_{jac}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1543: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1545: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p = \tilde{N}_{gs}^{-1} v$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1549: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}p_1 &  = & \tilde{A}_{11}^{-1} v_1  \nonumber\\ 
  p_2 &  = & \tilde{A}_{22}^{-1} (v_2 - A_{21} v_1 ) \label{eq-bceval} \\ 
  p_3 &  = & \tilde{A}_{33}^{-1} (v_3 - A_{31} v_1 - A_{32} v_2).\nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{ii}^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1551: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{ii} = L_i U_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1555: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ii}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1557: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(A_{ii})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1559: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ii}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1561: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(A_{ii}) r_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1563: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|\star \|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1565: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{i0} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1567: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1569: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_i = p(A_{ii}) v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1571: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(A_{ii})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1573: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{ii}^{-1} = p(A_{ii})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1575: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1579: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_k = <s_1, s_2, \ldots, s_k>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1581: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1583: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V_k = <v_1, v_2, \ldots, v_k>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1585: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$As_i = v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1587: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$As_i = v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1589: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| b - Ax_k \|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1591: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_k \in S_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1593: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\{ v_i \}_{i = 1, \ldots, k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1595: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1597: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_k \in S_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1599: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b - A x_k \perp v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1601: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i = 1, \ldots k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1603: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}(b-A x_k, v_i) = 0.  \label{eq-project-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Ax_k \in V_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1605: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Ax_k = \sum_{j = 1\ldots k} \lambda_j v_j \label{eq-project-2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_i = (b, v_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1607: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}A x_k = \sum_{i = 1\ldots k} (b,v_i) v_i.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$As_i = v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1609: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}A x_k = \sum_{i= 1\ldots k} (b,v_i) As_i = A \sum_{i=1\ldots k} (b,v_i) s_i
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_k = \sum_{i=1\ldots k} (b,v_i) s_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1611: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_{k+1} = x_k + (b, v_{k+1}) s_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1613: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_k = b - Ax_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1615: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_{k+1} = r_k -(b, v_{k+1}) v_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1617: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1619: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$As_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1621: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1623: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v_{k+1} = As_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1625: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1627: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1629: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1 \leq i \leq k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1631: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1633: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$As_i = v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1635: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}% latex2html id marker 560
[htb]
\begin{center}
\framebox[10.0cm]{
\parbox{10.0cm}{
\begin{tabbing}
\ \ \ \=\+$r_0 = b -Ax_0$; $k = 0$  \\ 
{\bf while}\= \ $\|r_k\| \geq \epsilon \|r_0\|$ \+ \\ 
  {\bf choose a search direction } $s_{k+1}$ \\ 
  {\bf compute} $v_{k+1} = As_{k+1}$ \\ 
  \# modified Gram-Schmidt \\ 
  {\bf for} \= $i = 1, \ldots, k$ \+\\ 
     $\alpha = (v_{k+1}, v_i)$ \\ 
     $v_{k+1} = v_{k+1} - \alpha \cdot v_i$ \\ 
     \# ensure $As_{k+1} = v_{k+1}$ \\ 
     $s_{k+1} = s_{k+1} - \alpha \cdot s_i$ \-\\ 
  {\bf end for} \\ 
  $\beta = \|v_{k+1}\|_2$ \\ 
  $v_{k+1} = v_{k+1} / \beta$ \\ 
  $s_{k+1} = s_{k+1} / \beta$ \\ 
  \# end Gram-Schmidt \\ 
  \# update $x$ and $r$ \\ 
  $\gamma = (b, v_{k+1})$ \\ 
  $x_{k+1} = x_{k+1} + \gamma s_{k+1}$ \\ 
  $r_{k+1} = r_{k+1} - \gamma s_{k+1}$ \\ 
  $k = k+1$ \-\\ 
{\bf end while } \\ 
\end{tabbing}
}}
\end{center}

\label{fig-gcr}         
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{k+1} = r_{k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1677: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1679: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$3/2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1681: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{k+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1683: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A s_{k+1} = r_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1685: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{k+1} = \tilde{N}^{-1} r_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1687: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{ii}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1689: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r = Q^T N^{-1}f - Q^T N^{-1}A Qv$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1691: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|r_k\| \leq \epsilon  \|r_0\|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1695: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}-\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} x^2} 
  -\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} y^2} = -4 \mbox{\ on \ } \Omega = [-1,1]\times [-1,1].  
       \label{eq-poisson}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u(x,y) = x^2+ y^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1701: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}-\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} x^2} 
  -\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} y^2} + a^1 \frac{\displaystyle \partial^{\,\!} u}{\displaystyle \partial_{\,\!} x} + a^2\frac{\displaystyle \partial^{\,\!} u}{\displaystyle \partial_{\,\!} y} 
  + 50 u = 1 \mbox{\ on \ } \Omega = [-1,1]\times [-1,1].  \label{eq-recirc}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left\{
\begin{array}{l}
  a^1(x,y) = 100\cdot y\cdot (1-x^2) \\ 
  a^2(x,y) = -100\cdot x\cdot (1-y^2)+10\cdot(y+1).
\end{array}
\right.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10\cdot (y+1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1703: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u(x,y) = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1705: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial u / \partial n
 = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1707: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}-\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} x^2} 
  -\frac{\displaystyle \partial^{\,\!} ^2u}{\displaystyle \partial_{\,\!} y^2} + 50 \frac{\displaystyle \partial^{\,\!} u}{\displaystyle \partial_{\,\!} x} + 50\frac{\displaystyle \partial^{\,\!} u}{\displaystyle \partial_{\,\!} y} 
  + 50 u = 2 \mbox{\ on \ } \Omega = [-1,1]\times [-1,1],  \label{eq-uniform}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P = L D^{-1}U$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1709: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left\{
\begin{array}{ll}
  l_{i,j} = a_{i,j} & \mbox{for \ } j < i \\ 
  u_{i,j} = a_{i,j} & \mbox{for \ } j > i \\ 
  d_{i} = u_{i,i} = l_{i,i} & \ .
\end{array}
\right.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(i,i-n_x), (i,i-1), (i,i), (i,i+1), (i,i+n_x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1717: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}d_{i} = a_{i,i} - a_{i,i-w} d^{-1}_{i-w} a_{i-w,i} 
                  - a_{i,i-1} d^{-1}_{i-1} a_{i-1,i} \mbox{\ for\ } 
                  i = 1,2,3,\ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_{i,j} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1719: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1723: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$ 1 \leq i \leq n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1725: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10^{-4}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1727: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mbox{itercount}/\mbox{time}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1735: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$80 \times 80$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1737: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$4\times 4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1739: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon = 10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1741: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon = 10^{-4}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1743: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10^{-4}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1745: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
  \begin{center}
  \begin{tabular}{||c|l|c|c|c||}
    \hline
    \ & $\epsilon$ &  {\em Poisson} & {\em Recirculating flow} & {\em Uniform flow} \\ 
    \hline
    I & $10^{-8}$      &  $19/38.74$  &  $12/20.74$ &  $7/7.98$   \\ 
    \ & $10^{-4}$      &  $19/22.98$  &  $12/13.52$ &  $7/5.60$    \\ 
    \hline
    II & $10^{-4}$           &  $14/16.91$  &  $9/9.91$  &  $5/3.74$    \\ 
    \  & $10^{-3}$           &  $14/14.00$  &  $9/8.68$  &  $5/3.30$    \\ 
    \  & $10^{-2}$           &  $15/12.27$  &  $9/7.38$   &  $5/2.82$    \\ 
    \  & $10^{-1}$           &  $17/9.90$  &  $10/6.47$  &  $6/2.93$    \\ 
    \hline
    III  & IBLUD post + GMRES  &  $33/4.43$   &  $46/6.24$  &  $16/2.13$   \\ 
    \  & IBLUD post + GCR    &  $33/5.19$   &  $46/7.49$  &  $16/2.50$   \\ 
    \hline
  \end{tabular}
  \end{center}
  
  \label{tab-lower}
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{-1}L^{-1}\times \mbox{vector}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1809: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1811: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u(x,y) = x^2+y^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1813: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|x\|_2 = \sqrt{\sum_{i=1, \ldots,n} x_i^2/n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1815: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
  \begin{tabular}{||c|l|c|c||}
    \hline
    \ & $\epsilon$ & $\| u^h-u\|_{\infty}$  & $\|u^h-u\|_2$  \\ 
    \hline
    I & $10^{-8}$      &  $0.00001$ & $0.0002$ \\ 
    \ & $10^{-4}$      &  $0.0020$  & $0.0030$ \\ 
    \hline 
    II & $10^{-4}$           &  $0.0032$    & $0.0016$ \\ 
    \  & $10^{-3}$           &  $0.0033$    & $0.0016$ \\ 
    \  & $10^{-2}$           &  $0.0029$    & $0.0014$ \\ 
    \  & $10^{-1}$           &  $0.0038$    & $0.0020$ \\ 
    \hline
    III  & IBLUD               &  $0.0010$    & $0.0022$ \\ 
    \hline
  \end{tabular}
\end{center}

\label{tab-resid}          
\end{table}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$80\times 80$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1869: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
  \begin{center}
  \begin{tabular}{||c|c|c|c||}
    \hline
    \ & {\em Poisson} & {\em Recirculating flow} & {\em Uniform flow} \\ 
    \hline
    ILUD preconditioner  & $39/3.74$   &  $52/5.04$    & $17/1.57$ \\ 
    ILUD postconditioner & $33/3.84$   &  $39/4.69$    & $16/1.85$ \\ 
    \hline
  \end{tabular}
  \end{center}
  
  \label{tab-single}
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b_x \times b_y - n_x\times n_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1883: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b_x \times b_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1885: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n_x \times n_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1887: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$2\times 2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1889: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$8\times 8=64$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1891: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon  = 10^{-4}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1893: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon =10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1895: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 756
W(m) = c\cdot m^{\alpha}  \label{eq-work}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1901
$\alpha > 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1901: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m = N/p$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1907: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 760
W = p\cdot W(N/p) = cN^{\alpha}\cdot \frac{1}{p^{\alpha-1}}. \label{eq-work2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ji}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1919: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1923: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10-20\%$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1925: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
\hline
  \ & \ & \multicolumn{3}{|c||}{decomposition}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2\times2 - 40\times 40$ & $4\times 4 -20\times20$ &
            $8\times8 - 10\times 10$ \\ 
\hline
I & $ 10^{-8}$              & $14/64.04$ &$19/38.74$ &$27/42.69$ \\ 
\ & $ 10^{-4}$              & $14/30.01$ &$19/22.98$ &$27/30.31$ \\ 
\hline
II & $  10^{-4}$             & $11/22.73$ &$14/16.91$ &$20/22.24$ \\ 
\  & $  10^{-3}$             & $11/18.73$ &$14/14.00$ &$21/21.36$ \\ 
\  & $  10^{-2}$             & $11/12.56$ &$15/12.27$ &$21/19.18$ \\ 
\  & $  10^{-1}$             & $13/8.64$ &$17/9.90$ &$24/19.44$ \\ 
\hline
III  &  IBLUD post + GMRES     & $33/3.85$ &$33/4.43$ &$35/5.82$ \\ 
\  & IBLUD post + GCR        & $33/4.73$ &$33/5.19$ &$35/7.00$ \\ 
 \hline
\end{tabular}
\end{center}

\label{tab-multi-poisson}         
\end{table}
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
\hline
 \multicolumn{5}{||c||}{Recirculating flow} \\ 
\hline
  \ & \ & \multicolumn{3}{|c||}{decomposition}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2\times2 - 40\times 40$ & $4\times 4 -20\times20$ &
            $8\times8 - 10\times 10$ \\ 
\hline
I & $ 10^{-8}$              & $8/24.80$ &$12/20.74$ &$20/30.07$ \\ 
\ & $ 10^{-4}$              & $8/14.38$ &$12/13.52$ &$20/23.61$ \\ 
\hline
II & $  10^{-4}$             & $6/9.99$ &$9/9.91$ &$16/18.16$ \\ 
\  & $  10^{-3}$             & $6/8.45$ &$9/8.68$ &$16/17.22$ \\ 
\  & $  10^{-2}$             & $7/7.64$ &$9/7.38$ &$16/15.95$ \\ 
\  & $  10^{-1}$             & $9/6.21$ &$10/6.47$ &$18/16.42$ \\ 
\hline
III  &  IBLUD post + GMRES     & $43/5.20$ &$46/6.24$ &$49/8.31$ \\ 
\  & IBLUD post + GCR        & $43/6.40$ &$46/7.49$ &$49/9.67$ \\ 
 \hline \hline
 \multicolumn{5}{||c||}{Uniform flow} \\ 
\hline 
  \ & \ & \multicolumn{3}{|c||}{decomposition}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2\times2 - 40\times 40$ & $4\times 4 -20\times20$ &
            $8\times8 - 10\times 10$ \\ 
\hline
I & $ 10^{-8}$              & $6/8.64$ &$7/7.98$ &$8/10.16$ \\ 
\ & $ 10^{-4}$              & $6/5.37$ &$7/5.60$ &$8/8.21$ \\ 
\hline
II & $  10^{-4}$             & $4/3.37$ &$5/3.74$ &$7/6.63$ \\ 
\  & $  10^{-3}$             & $4/2.97$ &$5/3.30$ &$7/6.17$ \\ 
\  & $  10^{-2}$             & $4/2.40$ &$5/2.82$ &$7/5.84$ \\ 
\  & $  10^{-1}$             & $5/2.02$ &$6/2.93$ &$8/5.99$ \\ 
\hline
III  &  IBLUD post + GMRES     & $16/1.89$ &$16/2.13$ &$16/2.74$ \\ 
\  & IBLUD post + GCR        & $16/2.29$ &$16/2.50$ &$16/3.05$ \\ 
 \hline 
\end{tabular}
\end{center}

\label{tab-multi-advdif}         
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ji}v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2135: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v = U^{-1}L^{-1} w$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2139: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ii}v$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2141: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{||l|c|c|c|c||}
  \hline
  \  & \multicolumn{4}{|c||}{\em Decomposition} \\  \cline{2-5} 
  {\em category} & $1\times 1$ & $2\times 2$ & $4\times 4$ & $8\times 8$ \\ 
  \hline
   {\em total time} & $3.81$ & $3.85$ & $4.43$ & $5.82$ \\ 
   \hline
   {\em copy}       & $0.33$ & $0.39$ & $0.71$ & $1.14$ \\ 
   {\em bc}         & $0.16$ & $0.23$ & $0.38$ & $0.88$ \\ 
   \hline
   {\em prec}       & $0.81$ & $0.75$ & $0.68$ & $0.66$ \\ 
   {\em matvec}     & $0.58$ & $0.57$ & $0.61$ & $0.64$ \\ 
  \hline
\end{tabular}
\end{center}

\label{tab-overhead}
\end{table}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2193: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2269: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$4\times 4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2271: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2269: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$4\times 4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2271: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{||c|c|c|c|c|c|c||}
 \hline
   \ & \multicolumn{2}{|c|}{\em Poisson} 
     & \multicolumn{2}{|c|}{\em Recirculating flow} 
     & \multicolumn{2}{|c||}{\em Uniform flow} \\ 
 \cline{2-7}
 $\eps$ & {\em seq} & {\em par} & {\em seq} & {\em par} & {\em seq} & {\em par} \\    
 \hline
 $10^{-4}$   & $14/16.91$ & $31/37.39$ & $9/9.91$  & $16/18.00$ & $5/3.74$ & $12/8.96$ \\ 
 $10^{-3}$   & $14/14.00$ & $32/31.95$ & $9/8.68$  & $16/15.99$ & $5/3.30$ & $12/8.00$ \\ 
 $10^{-2}$   & $15/12.27$ & $33/27.17$ & $9/7.38$  & $16/13.47$ & $5/2.82$ & $12/7.04$ \\ 
 $10^{-1}$   & $17/9.90$  & $34/19.94$ & $10/6.47$ & $17/11.34$ & $6/2.93$ & $12/5.97$ \\ 
 IBLUD+GMRES & $33/4.43$  & $44/5.85$  & $46/6.24$ & $53/6.91$ & $16/2.13$ & $21/2.81$ \\ 
\hline
\end{tabular}
\end{center}

\label{tab-par}         
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon  = 10^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2273: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2281: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}

\end{document}
