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{\newpage
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\samepage \begin{equation}V^{n+1} = V^{\star} - \Delta t G \Delta P.      \label{eq-p-3}
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{\newpage
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\samepage \begin{equation}S(V^n, P^n) = \frac{I}{\Delta t} - M(V^n, P^n)  \label{eq-momentum-eq}
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}

{\newpage
\clearpage
\samepage \begin{equation}T = D G              \label{eq-pressure-eq}
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{\newpage
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\samepage \begin{equation}A =
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{\newpage
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{\newpage
\clearpage
\samepage \begin{equation}S(V^n,P^n) = 
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    \begin{array}{ccc}
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      S_{21} & S_{22} & S_{23} \\ 
      S_{31} & S_{32} & S_{33} 
    \end{array}
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{\newpage
\clearpage
\samepage \begin{equation}\bar{S}(V^n, P^n)\bar{V}^{\star} = 
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  \begin{array}{cccc}
    S_{11} & S_{12} & 0      & S_{13} \\ 
    S_{21} & S_{22} & 0      & S_{23} \\ 
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    \bar{V}^{\star}_2 \\ 
    \bar{V'}^{\star}_2 \\ 
    \bar{V}^{\star}_3 
  \end{array}.
  \right] = 
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  \end{array}
  \right]                           \label{eq-equiv}
\end{equation}
}

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{\newpage
\clearpage
\samepage \begin{equation}A_{11} = {\left[
\begin{array}{cc}
S_{11} & S_{12} \\   S_{21} & S_{22} 
\end{array}
\right] }, \ \ \ 
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\begin{array}{cc}
S_{22} & S_{23} \\   S_{32} & S_{33} 
\end{array}
\right] }.
\end{equation}
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$v_1^{m+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2339: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{12}v_2^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2341: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$A_{12}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2343: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$III_{r}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2345: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$v_1^{m+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2347: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$v_2^{m+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2349: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$v_1^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2351: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$I_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2353: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$I_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2355: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$III_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2357: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$v_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2359: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v_{nr}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2361: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2367: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

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

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

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

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_{gs}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2371: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_{jac}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2373: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2377: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$I_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2379: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$III_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2381: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u = {\left[
\begin{array}{c}
 w \\   v
\end{array}
\right] }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2385: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$I_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2389: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$III_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2391: \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_inline2395: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Qv = {\left[
\begin{array}{c}
0 \\   v
\end{array}
\right] }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2397: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2399: \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.
  \label{eq-int-iter}
\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}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\tilde{N} u^{m+1} = (N - A) u^m + f  \label{eq-implement},
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(N-A)u^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2403: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N} u^{m+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2405: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$( A + \tilde{N} - N)u = f$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2407: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2411: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}u^{m+1} = u^m + \tilde{N}^{-1}( f - A u^m),  \label{eq-approx}
\end{equation}
}

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

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

        \tilde{A}_{11} & \emptyset  \\  
        \emptyset & \tilde{A}_{22} \\  
      
\end{array}
\right] }                     .                  \label{eq-tilde-N}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}_{gs}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2421: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}_{jac}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2423: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2425: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{ii}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2427: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p = \tilde{N}_{gs}^{-1} t$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2429: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}p_1 &  = & \tilde{A}_{11}^{-1} t_1  \nonumber\\ 
  p_2 &  = & \tilde{A}_{22}^{-1} (t_2 - A_{21} t_1 ) \label{eq-bceval} 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{11}^{-1}t_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2431: \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_inline2435: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ii}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2437: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(A_{ii})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2439: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{ii}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2441: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(A_{ii}) r_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2443: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{i0} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2445: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$v_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2447: \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_inline2449: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(A_{ii})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2451: \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_inline2453: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2457: \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_inline2459: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A\tilde{N}^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2461: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\sqrt\kappa -1 }{\sqrt\kappa + 1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2463: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}N u = g  \label{eq-N}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\tilde{N} \tilde{u} = g \label{eq-N-2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2467: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{condition}\label{cond-relacc}
  Each subproblem $A_{ii} u_i = g_i$ is solved using initial guess $0$
  and with a relative accuracy of $\eps$ so that 
  $\| g_i - A_{ii} \tilde{u}_i\| \leq \epsilon  \| g_i \|$
  in the Euclidean norm. 
\end{condition}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2479: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{theorem}% latex2html id marker 755
\label{th-relacc}
  If condition~\ref{cond-relacc} holds for all subdomains and all possible
  right-hand sides $g_i$, then
  \begin{itemize}
   \item[(a)] $\| I - N_{gs}\tilde{N}^{-1}_{gs} \| \leq C \eps$,
   for some constant $C > 0$. \\ 
  \item[(b)] $\| I - N_{jac}\tilde{N}^{-1}_{jac} \| \leq \eps$.
  \end{itemize}
\end{theorem}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}_{ii} \tilde{u_i} = g_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2489: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| g_i-A_{ii}\tilde{A}_{ii}^{-1}g_i\| = 
  \| (I- A_{ii}\tilde{A}_{ii}^{-1}) g_i \| \leq \epsilon  \| g_i \|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2491: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2493: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| I - A_{ii}\tilde{A}_{ii}^{-1}\| \leq \eps$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2495: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2499: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}I -N\tilde{N}^{-1} = 
    {\left[
\begin{array}{cc}
 I-A_{11}\tilde{A}_{11}^{-1} & \emptyset \\  
                 -(I-A_{22}\tilde{A}_{22}^{-1})A_{21}\tilde{A}_{11}^{-1} &
                     I-A_{22}\tilde{A}_{22}^{-1} 
\end{array}
\right] }
  \end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x = {\left[
\begin{array}{c}
x_1 \\   x_2
\end{array}
\right] }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2501: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| x \| \leq \left\| {\left[
\begin{array}{c}
x_1\\   0 
\end{array}
\right] } \right\| + \left\| {\left[
\begin{array}{c}
 0 
   \\   x_2
\end{array}
\right] }\right\|
  = \| x_1 \| + \| x_2 \|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2503: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$
    \|I -N\tilde{N}^{-1}\| = \sup_{\|x\| \leq 1}
        \|(I -N\tilde{N}^{-1})x\|  \leq 
      \sup_{\|x\| \leq 1 } \left( \|(I-A_{11}\tilde{A}_{11}^{-1})x_1\| 
      + \|(I-A_{22}\tilde{A}_{22}^{-1})A_{21}\tilde{A}_{11}^{-1}x_1\| +
         \| (I-A_{22}\tilde{A}_{22}^{-1} )x_2\| \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2505: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|x\| \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2507: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|x_1\| \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2509: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|x_2\| \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2511: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C = 2 + \|A_{21}\tilde{A}_{11}^{-1}\|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2513: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B = \mbox{diag}(D_1, D_2, \ldots, D_n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2515: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| B\| = \sqrt{\rho(B^TB)} = 
     \sqrt{ 
        \rho(\mbox{diag}(D_1^TD_1, \ldots, D_n^TD_n)) } =
        \max \{ \sqrt{\rho(D_1^TD_1)}, \ldots, \sqrt{\rho(D_n^TD_n)} \} =
        \max \{ \|D_1\|, \ldots \|D_n \| \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2517: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$I-N\tilde{N}^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2519: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D_i = I - A_{ii}\tilde{A}_{ii}^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2521: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| I - N\tilde{N}^{-1} \| = \max_{i}{\|I - A_{ii}\tilde{A}_{ii}^{-1}\|} 
    \leq \eps$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2523: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Box$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2525: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A\tilde{N}^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2527: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$AN^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2529: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{theorem}% latex2html id marker 861
\label{th-upperbound}
  Under the conditions of Theorem~\ref{th-relacc} and $C\epsilon  < 1$, 
  the condition number
  of $A\tilde{N}^{-1}$ satisfies
  \begin{equation}
    \kappa(A\tilde{N}^{-1}) \leq 
      \frac { 1  + C \epsilon }{1 - C\epsilon }\cdot \kappa(AN^{-1}). \label{eq-kappa}
  \end{equation}
\end{theorem}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| \star \|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2535: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| N\tilde{N}^{-1} \| = \|N\tilde{N}^{-1} - I + I \| \leq 1 + C\eps$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2537: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\| (N\tilde{N}^{-1})^{-1} \| = \| (N\tilde{N}^{-1})^{-1} (I - N\tilde{N}^{-1})
        + I\| \leq 1 + \| (N\tilde{N}^{-1})^{-1} \| C\eps$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2539: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C\epsilon  < 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2541: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa(N\tilde{N}^{-1}) = \| N\tilde{N}^{-1} \|\cdot \|(N\tilde{N}^{-1})^{-1}\|
    \leq \frac { 1  + C \epsilon }{1 - C\epsilon }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2543: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa(A\tilde{N}^{-1}) = 
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[htb]
\begin{center}
\framebox[10.0cm]{
\parbox{10.0cm}{
\begin{tabbing}
\ \ \ \=\+$r_0 = f -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 $r$ \\ 
  $\gamma = (f, v_{k+1})$ \\ 
  $x_{k+1} = x_{k+1} + \gamma s_{k+1}$ \\ 
  $r_{k+1} = r_{k+1} - \gamma s_{k+1}$ \\ 
  $k = k+1$ \-\\ 
{\bf end while } \\ 
\end{tabbing}
}}
\end{center}

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

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\centerline{
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    \psfig{file=cyl.eps,width=6.0cm,height=3.0cm} &
    \psfig{file=gridmulti.eps,width=6.0cm,height=3.0cm} \\ 
    (a) & (b)  \\ 
    \psfig{file=gridsingle.eps,width=6.0cm,height=3.0cm} & \ \\ 
    (c) & \ \\ 
  \end{tabular}
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\label{fig-geom}
\end{figure}
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    \psfig{file=smulti.eps,width=12.0cm,height=6.0cm,angle=-90} 
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\label{fig-s}
\end{figure}
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\samepage \begin{table}% latex2html id marker 1112
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  \begin{center}
  \begin{tabular}{||c|l|c|c|c|c||}
    \hline
    \ & $\epsilon$ &  {\em Total} & {\em Momentum} & {\em Pressure} & {\em Other} \\ 
    \hline
    I & $10^{-8}$      &  $924.7$ &$146.6(37)$ &$722.8(157)$ &$50.5$ \\ 
    \ & $10^{-4}$      &  $449.1$ &$78.6(38)$ &$315.2(154)$ &$50.5$ \\ 
    \hline
    II & $10^{-4}$     &  $465.2$ &$58.4(37)$ &$351.6(155)$ &$50.5$ \\ 
    \  & $10^{-2}$     &  $292.1$ &$43.5(38)$ &$193.4(168)$ &$50.5$ \\ 
    \  & $10^{-1}$     &  $230.3$ &$47.5(48)$ &$127.6(210)$ &$50.4$ \\ 
    \hline
    III  &     RIBLU($\alpha$) post + GCR  (trunc)  &  
           $190.3$ &$28.4(85)$ &$106.8(566)$ &$50.4$ \\ 
    \  &  RIBLU($\alpha$) post + GCR (restart)  & $179.0$ & 
               $26.7(85)$ & $97.0(646)$ & $50.6$ \\ 
    \hline
  \end{tabular}
  \end{center}
  
  \label{tab-lower}
\end{table}
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{\newpage
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\samepage \begin{table}[htb]
  \begin{center}
  \begin{tabular}{||c|c|c|c||}
    \hline
    \multicolumn{4}{||c||}{\em Poisseuille flow} \\ 
    \hline
    {\em Total} & {\em Momentum} & {\em Pressure} & {\em Other} \\ 
    \hline
    $94.0$ & $46.5(407)$ & $21.0(395)$ & $24.4$ \\ 
    \hline
    \hline
    \multicolumn{4}{||c||}{\em Cylinder problem} \\ 
    \hline
    {\em Total} & {\em Momentum} & {\em Pressure} & {\em Other} \\ 
    \hline
    $128.0$ & $31.6(140)$ & $48.9(497)$ & $44.0$ \\     
    \hline
  \end{tabular}
  \end{center}
  
  \label{tab-single}
\end{table}
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
  \hline
  \multicolumn{5}{||c||}{Poisseuille flow} \\ 
  \hline
  \ & \ & \multicolumn{3}{|c||}{decomposition}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2\times2$ & $4\times 4$ & $5\times5$ \\ 
\hline
I & $ 10^{-8}$              & $333.6(101)$ & $190.8(120)$ & $171.3(129)$  \\ 
\ & $ 10^{-4}$              & $147.3(102)$ &   $90.5(121)$ &   $87.5(129)$ \\ 
\hline
II & $  10^{-4}$             & $120.1( 89)$ &   $81.0(106)$ &  $82.8(116)$ \\ 
\  & $  10^{-2}$             & $55.6( 89)$ &   $50.6(108)$ &   $56.6(117)$ \\ 
\  & $  10^{-1}$             & $45.3(111)$ &   $49.9(136)$ &   $57.1(146)$ \\ 
\hline
III  &  RIBLUD($0.95$) post + GCR (trunc)    & $51.5(241)$ &   $78.8(281)$ &  
        $92.5(295)$ \\ 
\  & RIBLUD($0.95$) post + GCR (restart)     & $42.8(241)$ &   $65.8(281)$ &   $86.4(295)$ \\ 
 \hline
  \hline
  \multicolumn{5}{||c||}{Cylinder problem} \\ 
  \hline
  \ & \ & \multicolumn{3}{|c||}{no. of blocks}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2$ & $4$ & $8$ \\ 
\hline
I & $ 10^{-8}$              & $146.6(37)$ &   $145.9(49)$ &   $113.6(49)$ \\ 
\ & $ 10^{-4}$              & $78.6(38)$ &   $80.7(49)$ &   $63.0(49)$ \\ 
\hline
II & $  10^{-4}$             & $58.4(37)$ &   $65.7(47)$ &   $53.0(47)$ \\ 
\  & $  10^{-2}$             & $43.5(38)$ &   $49.3(47)$ &   $42.4(47)$ \\ 
\  & $  10^{-1}$             & $47.5(48)$ &   $52.5(56)$ &   $59.8(56)$ \\ 
\hline
III  &  RIBLUD($0.95$) post + GCR (trunc)    & $28.4(85)$ &   $30.8(86)$ &   $30.8(86)$ \\ 
\  & RIBLUD($0.95$) post + GCR (restart)     & $26.7(85)$ &   $29.1(86)$ &   $29.0(86)$ \\ 
 \hline 
\end{tabular}
\end{center}

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

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[htb]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
  \hline
  \multicolumn{5}{||c||}{Poisseuille flow} \\ 
  \hline
  \ & \ & \multicolumn{3}{|c||}{decomposition}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2\times2$ & $4\times 4$ & $5\times5$ \\ 
\hline
I & $ 10^{-8}$              & $135.6(180)$ & $104.2(297)$ & $120.5(334)$ \\ 
\ & $ 10^{-4}$              & $63.8(188)$ &   $59.7(303)$ &   $77.1(344)$ \\ 
\hline
II & $  10^{-4}$             & $77.8(201)$ &   $85.2(323)$ &   $108.2(361)$ \\ 
\  & $  10^{-2}$             & $52.9(216)$ &   $69.1(333)$ &   $90.8(374)$ \\ 
\  & $  10^{-1}$             & $56.0(303)$ &   $81.5(422)$ &   $103.8(456)$ \\ 
\hline
III  &  RIBLU($\alpha$) post + GCR (trunc)    & $53.7(483)$ &   $100.6(633)$ &  
$120.3(660)$ \\ 
\  & RIBLU($\alpha$) post + GCR (restart)     & $43.3(599)$ &   $84.8(767)$ &  
$124.4(867)$ \\ 
 \hline
  \hline
  \multicolumn{5}{||c||}{Cylinder problem} \\ 
  \hline
  \ & \ & \multicolumn{3}{|c||}{no. of blocks}\\ 
  \cline{3-5}
  \ & $\epsilon$ & $2$ & $4$ & $8$ \\ 
\hline
I & $ 10^{-8}$              & $722.8(157)$ &   $688.9(282)$ &   $496.2(343)$ \\ 
\ & $ 10^{-4}$              & $315.2(154)$ &   $328.4(285)$ &   $247.7(344)$ \\ 
\hline
II & $  10^{-4}$             & $351.6(155)$ &   $352.1(279)$ &   $283.2(339)$ \\ 
\  & $  10^{-2}$             & $193.4(168)$ &   $208.6(296)$ &   $201.6(374)$ \\ 
\  & $  10^{-1}$             & $127.6(210)$ &   $155.1(371)$ &   $176.6(467)$ \\ 
\hline
III  &  RIBLU($\alpha$) post + GCR (trunc)    & $106.8(566)$ &   $149.5(742)$ &   $183.0(811)$ \\ 
\    & RIBLU($\alpha$) post + GCR (restart)   & $97.0(646)$  &   $173.2(1076)$ &  $225.2(1332)$ \\ 
 \hline 
\end{tabular}
\end{center}

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

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{\newpage
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\samepage \begin{figure}[htb]
\centerline{
\begin{tabular}{cc}
    \psfig{file=momalpha.eps,width=6.0cm,height=4.9cm} &
    \psfig{file=palpha.eps,width=6.0cm,height=4.9cm} \\ 
    (a) & (b) \\ 
\end{tabular}
}

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

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\begin{center}
\begin{tabular}{||c|c|c||c|c||}
 \hline
   \ & \multicolumn{2}{|c||}{\em Multiplicative}  & 
       \multicolumn{2}{|c|}{\em Additive} \\ 
 \cline{2-5}
 $\eps$ & {\em momentum} & {\em pressure} & {\em momentum} & {\em pressure} \\    
 \hline
 $10^{-4}$   & $53.0(47)$ & $283.2(339)$ & $87.0( 76)$ & $574.8(674)$ \\ 
 $10^{-2}$   & $42.4(47)$ & $201.6(374)$ & $68.2( 76)$ & $394.9(749)$ \\ 
 $10^{-1}$   & $59.8(56)$ & $176.6(467)$ & $68.1( 84)$ & $276.0(736)$ \\ 
 RIBLU($\alpha$)+GCR (trunc)   & $30.8(86)$  & $183.0(811)$  & $37.0(101)$ & $213.2(998)$ \\ 
 RIBLU($\alpha$+GCR (restart) & $29.0(86)$  & $225.2(1332)$ & $34.5(101)$ & $332.3(1965)$ \\ 
\hline
\end{tabular}
\end{center}

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

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