\documentclass[letterpaper,12pt]{article}
\usepackage[cm,nobanner,notexpower,marsdefs,2002IA]{marslide}
%\usepackage[cm,nobanner,notexpower,marsdefs]{marslide}
\begin{document}
%\Draft
%\SetVersionDate{October 18, 2002}

\addtolength{\parskip}{0.25in}
\newcommand{\dlp}{\ensuremath{\delta p}}
\newcommand{\dlpbar}{\ensuremath{\delta \bar{p}}}
\newcommand{\dlptwidi}{\ensuremath{\delta \tilde{p}_i}}
\newcommand{\dlphatj}{\ensuremath{\delta \hat{p}_j}}
\newcommand{\dlw}{\ensuremath{\delta w}}
\newcommand{\dlW}{\ensuremath{\delta W}}
\newcommand{\dlu}{\ensuremath{\delta \mathbf{u}}}
\newcommand{\dlubar}{\ensuremath{\delta \mathbf{\bar{u}}}}
\newcommand{\dlutwidi}{\ensuremath{\delta \mathbf{\tilde{u}}_i}}
\newcommand{\dluhatj}{\ensuremath{\delta \mathbf{\hat{u}}_j}}
\newcommand{\dlv}{\ensuremath{\delta \mathbf{v}}}
\newcommand{\dlV}{\ensuremath{\delta \mathbf{V}}}
\newcommand{\pc}{\ensuremath{p_c}}
\newcommand{\wc}{\ensuremath{w_c}}
\newcommand{\Wc}{\ensuremath{W_c}}
\newcommand{\uc}{\ensuremath{\mathbf{u}_c}}
\newcommand{\vc}{\ensuremath{\mathbf{v}_c}}
\newcommand{\Vc}{\ensuremath{\mathbf{V}_c}}
\newcommand{\vgn}{\ensuremath{\mathbf{v}_{g_N}}}
\newcommand{\divdot}{\ensuremath{\mathbf{\nabla} \cdot}}
\newcommand{\Wh}{\ensuremath{W_h}}
\newcommand{\Vh}{\ensuremath{\mathbf{V}_h}}
\newcommand{\WHh}{\ensuremath{W_{H,h}}}
\newcommand{\VHh}{\ensuremath{\mathbf{V}_{H,h}}}
\newcommand{\WH}{\ensuremath{W_H}}
\newcommand{\VH}{\ensuremath{\mathbf{V}_H}}
\newcommand{\dlWh}{\ensuremath{\delta W_h}}
\newcommand{\dlVh}{\ensuremath{\delta \mathbf{V}_h}}
\newcommand{\pHh}{\ensuremath{p_{H,h}}}
\newcommand{\uHh}{\ensuremath{\mathbf{u}_{H,h}}}

% ----------------------------------------------------------------------
% Title Page
% ----------------------------------------------------------------------

%\null\vskip0.5in \vfill
\vfill
\begin{center}
   \frontpagetitle{Subgrid Upscaling Preconditioners for Darcy Flow}
   \vskip 0.75in
   \frontpageauthor{James M. Rath}
   \vspace{-0.5in}
   \frontpageaffil{Center for Subsurface Modeling / TICAM}
   \vspace{-0.5in}
   \frontpageaffil{Ph.~D. Advisor: Todd Arbogast}
\end{center}
\vfill

%\centerline{\lower 0pt\hbox{\LeftLogo}\hfill\lower 0pt\hbox{\RightLogo}}
\frontpagelogos

\newpage

% ----------------------------------------------------------------------
% Page 1: Problem statement and idea for resolution
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Problem:}


%FIX ME: change color of the equations
\begin{center}
{\LARGE \textcolor{bo}{$\divdot \mathbf{u} = q$} }

\vspace{0.5in}

{\LARGE \textcolor{bo}{$\mathbf{u} = -\frac{\mathbf{\kappa}}{\mu} (\mathbf{\nabla} p - \rho \mathbf{g})$} }
\end{center}


{\small
Approximate flow equations using a cell-centered finite difference method (CCFD)
for the pressures.
}

\vspace{0.5in}
\textcolor{greyblue}{\LARGE Difficulties:}

{\small
Heterogeneity in the permeability and its fine-scale resolution makes the
problem ill-conditioned and computationally expensive to solve.
}


\newpage

\textcolor{greyblue}{\LARGE The Problem in Mixed Variational Form:}

{\Large
\hspace{0.25in} Let

\begin{center}
\begin{tabular}{l l l}
$\mathbf{V}$ & $=$ & $\{ \mathbf{v} \in H(\mathrm{div},\Omega) \; | \;
\mathbf{v} \cdot \nu = 0 \;\mathrm{on}\; \partial \Omega \}$, \\
$W$ & $=$ & $L^2(\Omega)$. \\
\end{tabular}
\end{center}

\hspace{0.25in} Find $\mathbf{u} \in \mathbf{V}$ and $p \in W$ such that

\begin{center}
\begin{tabular}{l l}
$(\divdot \mathbf{u},w) = (q,w)$ & $\forall w \in W$, \\
$(\mu \mathbf{\kappa}^{-1} \mathbf{u}, \mathbf{v}) - (p,\divdot \mathbf{v}) =
(\rho \mathbf{g}, \mathbf{v}) \;\;$ & $\forall \mathbf{v} \in \mathbf{V}$. \\
\end{tabular}
\end{center}
}

\newpage

\textcolor{greyblue}{\LARGE The Finite Element Approximation:}

For a fine-scale problem, assume $\mathbf{\kappa}$ is piecewise constant on a
grid with spacing $h$.

Find $\mathbf{u}_h \in \Vh \subset \mathbf{V}$ and
$p_h \in \Wh \subset W$ such that

\begin{center}
\begin{tabular}{l l}
$(\divdot \mathbf{u}_h,w_h) = (q,w_h)$ & $\forall w_h \in \Wh$, \\
$(\mu \mathbf{\kappa}^{-1} \mathbf{u}_h, \mathbf{v}_h) - (p_h,\divdot \mathbf{v}_h) =
(\rho \mathbf{g}, \mathbf{v}_h) \;\;$ & $\forall \mathbf{v}_h \in \Vh$. \\
\end{tabular}
\end{center}

where $\Wh \times \Vh$ are Raviart-Thomas-Nedelec elements of order zero (RT0)
on the given grid.

Using quadrature --- the trapezoidal rule --- to compute the
$(\mu \mathbf{\kappa}^{-1} \mathbf{u}_h, \mathbf{v}_h)$ term is equivalent to
solving the original problem using CCFD.

\newpage

\begin{center}
Degrees of freedom for 2-D RT0 elements on a {\large
\textcolor{orangered}{fine}} rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,4){13}{ \line(1,0){72} }
\multiput(0,0)(6,0){13}{ \line(0,1){48} }
\textcolor{seafoamgreen}{
%  \multiput(3,0)(6,0){12}{\multiput(0,2)(0,4){12}{\makebox(0,0){\circle*{1}}}}
%  \multiput(3,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
%  \multiput(6,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
  \multiput(4.25,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
}
\textcolor{orangered}{
  \multiput(0,0)(6,0){13}{\multiput(0,2)(0,4){12}{\makebox(0,0){$\times$}}}
  \multiput(0,0)(0,4){13}{\multiput(3,0)(6,0){12}{\makebox(0,0){$\times$}}}
}
\end{picture}}
\end{center}

\bigskip

\centerline{ \textcolor{orangered}{$\times$} Velocity }

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

\newpage

\vspace*{0.25in}
If instead we desire a coarse-scale approximation, we can find
$\mathbf{u}_H \in \VH \subset \mathbf{V}$ and $p_H \in \WH \subset W$ such that

\begin{center}
\begin{tabular}{l l}
$(\divdot \mathbf{u}_H,w_H) = (q,w_H)$ & $\forall w_H \in \WH$, \\
$(\mu \mathbf{\kappa}^{-1} \mathbf{u}_H, \mathbf{v}_H) - (p_H,\divdot \mathbf{v}_H) =
(\rho \mathbf{g}, \mathbf{v}_H) \;\;$ & $\forall \mathbf{v}_H \in \VH$. \\
\end{tabular}
\end{center}

where $\WH \times \VH$ are RT0 elements (or perhaps
Brezzi-Douglas-Dur\`{a}n-Fortin order one (BDDF1) elements) on a grid with
spacing $H = C h$.

With the RT0 elements, this is equivalent to using a
$\mathbf{\kappa}$ which is piecewise constant on the $H$ grid, and is the
harmonic average of the fine-scale $\mathbf{\kappa}$.

\newpage

\begin{center}
Degrees of freedom for 2-D BDDF1 elements on a {\large
\textcolor{orangered}{coarse}} rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,16){4}{ \line(1,0){72} }
\multiput(0,0)(24,0){4}{ \line(0,1){48} }
\textcolor{seafoamgreen}{
  \multiput(13.5,0)(24,0){3}{\multiput(0,8)(0,16){3}{\makebox(0,0){\circle*{1}}}}
}
\textcolor{orangered}{
%  \multiput(0,0)(24,0){4}{\multiput(0,8)(0,16){3}{\makebox(0,0){$\otimes$}}}
%  \multiput(0,0)(0,16){4}{\multiput(12,0)(24,0){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(24,0){4}{\multiput(0,4)(0,16){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(24,0){4}{\multiput(0,12)(0,16){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(0,16){4}{\multiput(6,0)(24,0){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(0,16){4}{\multiput(18,0)(24,0){3}{\makebox(0,0){$\otimes$}}}
}
\end{picture}}
\end{center}

\bigskip

\centerline{ \textcolor{orangered}{$\otimes$} Velocity (linear) }

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

\newpage

\vspace*{0.25in}
\textcolor{greyblue}{\LARGE Idea:}

{\small
Use a multiscale approximation as the ``coarse'' level in a two-level iterative
scheme to precondition a fine-scale CCFD approximation.  By upscaling the
original problem (rather than simply coarsening it as in multigrid (MG)), we
retain more of the fine scale information in the computation thereby (hopefully)
obtaining a better approximation and an overall faster scheme.
}

{\small
Our subgrid upscaling technique uses the mixed finite element method (MFEM)
with a variational multiscale technique.
}


\newpage

% ----------------------------------------------------------------------
% Pages 2-?: Description of the upscaling technique
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE The Upscaling Technique:}

{\small
\textcolor{bo}{Decompose} $\mathbf{V}$ and $W$ (using a chosen coarse grid)
so that:
\vspace{-0.25in}
\begin{enumerate}
\item[(1)] we do not lose any information, $\mathbf{V} = \Vc \oplus \dlV$, $W =
\Wc \oplus \dlW$;
\item[(2)] mass is conserved, $\divdot \Vc = \Wc$, $\divdot \dlV = \dlW$;
\item[(3)] the fine-scale velocities are locally supported over the coarse grid,
$\dlV \cdot \nu = 0$ on the boundary of each coarse cell.
\end{enumerate}
}

%\vspace*{0.25in}
{\small
Condition (3) allows us to \textcolor{bo}{disconnect} the subgrid problems from
one another --- but not from the coarse grid problem.  We employ a Green's or
\textcolor{bo}{influence function} approach to effect a \textcolor{bo}{forward
elimination} of the subgrid problems into the coarse problem, and --- once the
coarse problem has been solved --- a \textcolor{bo}{back substitution} to
recover the subgrid solutions.  The two can then be recombined to obtain a fine
scale solution.
}


\newpage
%\vfill

\textcolor{greyblue}{\LARGE Summary of Advantages:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Each subgrid problem is independent from the others; they can be solved in
parallel with no communication needed between coarse cells.
\item Each subgrid
problem also has many fewer degrees of freedom than the whole problem (by about
a factor of the number of coarse grid cells).
\item Each subgrid problem has the same LHS and many different RHS creating an
opportunity for efficiency.
\item The coarse problem has many fewer degrees of freedom than the whole problem
(by about a factor of the number of fine cells chosen per coarse cell).
\item If the coarse grid spacing is a small multiple of the fine grid spacing,
the work in solving the upscaled problem is nearly all done in solving the
coarse problem.
\item There is only one coarse grid (not many as in MG).
\end{list}
}


\newpage

\textcolor{greyblue}{\LARGE Approximation of the scale-separated problem:}

{\small
We use Raviart-Thomas-Nedelec order zero elements to approximate $\dlW$ and
$\dlV$, denoted $\dlWh$ and $\dlVh$, and BDDF order one elements for $\Wc$ and
$\Vc$, denoted $\WH$ and $\VH$.

\begin{description}
\item[\textcolor{bo}{$\WH$}:] functions which are piecewise constant
on coarse cells
\item[\textcolor{bo}{$\dlWh$}:] functions which are piecewise constant
on fine cells, but must have a zero average over each coarse cell
\item[\textcolor{bo}{$\VH$}:] vector-valued functions where the components
normal to coarse cell faces are continuous across those faces, and vary linearly
along those faces
\item[\textcolor{bo}{$\dlVh$}:] vector-valued functions
where the components normal to fine cell faces are continuous across those
faces, and are constant along those faces; the normal components
across coarse cell faces are zero
\end{description}

\newpage

\begin{center}
Degrees of freedom for 2-D RT0 elements on a {\large
\textcolor{orangered}{fine}} rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,4){13}{ \line(1,0){72} }
\multiput(0,0)(6,0){13}{ \line(0,1){48} }
\textcolor{seafoamgreen}{
  \multiput(4.25,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
}
\textcolor{orangered}{
  \multiput(0,0)(6,0){13}{\multiput(0,2)(0,4){12}{\makebox(0,0){$\times$}}}
  \multiput(0,0)(0,4){13}{\multiput(3,0)(6,0){12}{\makebox(0,0){$\times$}}}
}
\end{picture}}
\end{center}

\bigskip

\centerline{ \textcolor{orangered}{$\times$} Velocity }

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

\newpage

%color diagram from slide 28 of Xian talk
\begin{center}
Degrees of freedom for 2-D RT0/BDDF1 elements on an ``{\large
\textcolor{orangered}{upscaled}}'' rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,16){3}{\multiput(0,0)(24,0){3}{
\textcolor{orangered}{%\linethickness{0.75pt}
\put(0, 4){\line(1,0){24}}\multiput(3, 4)(6,0){4}{\makebox(0,0){{$\times$}}}
\put(0, 8){\line(1,0){24}}\multiput(3, 8)(6,0){4}{\makebox(0,0){{$\times$}}}
\put(0,12){\line(1,0){24}}\multiput(3,12)(6,0){4}{\makebox(0,0){{$\times$}}}
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\put(18,0){\line(0,1){16}}\multiput(18,2)(0,4){4}{\makebox(0,0){{$\times$}}}
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\textcolor{seafoamgreen}{\multiput(2.5,0)(6,0){4}{\multiput(0,2)(0,4){4}{\makebox(0,0){\circle*{1}}}}}
\put(-1,0){
\put( 0,3){\makebox(0,0){{$\otimes$}}}\put( 0,13){\makebox(0,0){{$\otimes$}}}
\put(24,3){\makebox(0,0){{$\otimes$}}}\put(24,13){\makebox(0,0){{$\otimes$}}}
\put(5, 0){\makebox(0,0){{$\otimes$}}}\put(19.5, 0){\makebox(0,0){{$\otimes$}}}
\put(5,16){\makebox(0,0){{$\otimes$}}}\put(19.5,16){\makebox(0,0){{$\otimes$}}}
{\linethickness{1.5pt}\framebox(24,16){}}}
}}
\end{picture}}
\end{center}

\bigskip

\centerline{
$\otimes$ Coarse velocity (linear)\qquad
\textcolor{orangered}{$\times$} Subgrid velocity}

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

%\vspace{0.25in}
%\centerline{\setlength\unitlength{1pt}
%\begin{picture}(300,160)(0,0)
%  \multiput(0,0)(0,80){2}{
%  \multiput(0,0)(100,0){3}{
%    \thicklines
%    \put(0,0){\framebox(100,80){}}
%    \thinlines
%    \multiput(0,0)(100,0){2}{
%      \put(0,26){\makebox(0,0){\tiny $\otimes$}}
%      \put(0,54){\makebox(0,0){\tiny $\otimes$}}
%    }
%    \multiput(0,0)(0,80){2}{
%      \put(33,0){\makebox(0,0){\tiny $\otimes$}}
%      \put(67,0){\makebox(0,0){\tiny $\otimes$}}
%    }
%    \thinlines
%    \put(20,0){\framebox(20,80){}}
%    \put(60,0){\framebox(20,80){}}
%    \put(0,20){\framebox(100,20){}}
%    \put(0,60){\framebox(100,20){}}
%    \multiput(20,0)(20,0){4}
%      {\multiput(0,10)(0,20){4}{\makebox(0,0){\tiny $\times$}}}
%    \multiput(0,20)(0,20){3}
%      {\multiput(10,0)(20,0){5}{\makebox(0,0){\tiny $\times$}}}
%    \multiput(10,0)(20,0){5}{\multiput(0,10)(0,20){4}{\circle*{2}}}
%  }}
%\end{picture}}
%\medskip
%\centerline{\parbox{6.5in}{\tiny Figure~1 from~\cite{Arbogast_2002}.
%A 2-D mesh of $15\times8$ fine grid elements decomposed
%into a $3\times2$ coarse grid with $5\times4$ subgrid elements per
%coarse element.  The dots represent the pressure values, one per fine
%element.  The crosses represent the edge velocity fluxes.  The circled
%crosses apply to the coarse grid edges, and represent linear flux
%variation.  The other crosses represent constant fluxes across the
%subgrid edges internal to the coarse elements.}}

%\null\vfill
%    \begin{center}
%%  \includegraphics[scale=1.5,angle=0]{graphic}
%    \vspace{1.5in}  \fbox{image}\\
%    \bigskip\bigskip
%    \TextCcaption{%
%        Caption $x =\frac1{2a}\left(-b\pm\sqrt{b^2-4ac}\right)$.}
%    \label{fig1}
%    \end{center}
%\vfill\null
}


\newpage

\textcolor{greyblue}{\LARGE Error Estimates for the Approximation:}

{\small
No closure assumption is made regarding the permeability.

Accuracy is simply a question of approximation
theory --- how well does $\VHh$ approximate $\mathbf{V}$? --- and not how much
the physics being changed.

Our closure assumption can be said to be that all flux across a
coarse element face is due to the coarse scale functions.  To
compensate for the coarseness of this restriction, higher order
accurate basis functions have been employed.

For the BDDF1/RT0 elements:

\begin{tabular}{l l}
\hspace{0.5in} &
$\|\mathbf{u} - \uHh\|_0 \le C\|\mathbf{u}\|_2 H^2 = O(H^2)$ \\
\hspace{0.5in} &
$\|p - \pHh\|_0 \le
C\{[\|p\|_1 + \|\divdot\mathbf{u}\|_1 h] h + \|\mathbf{u}\|_2 H^3\} =
O(h + H^3)$ \\
\end{tabular}
}


% ----------------------------------------------------------------------
% Pages 2-?: Some results for the upscaling technique
% ----------------------------------------------------------------------

\newpage

\textcolor{greyblue}{\LARGE Some Practical Results:}

\begin{center}
{\large Quarter Five-spot Oil Reservoir Water Flood }

Logarithm of the Permeability Field
\end{center}

\vspace{-0.5in}
\begin{center}
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=4.5in]{log10permx}

{\small Fine $40\times40$}
\end{center}

\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c c}

\multicolumn{3}{c}{\large Water saturation contours at 100 days} \\

\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{s1.051.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{s0.048.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{s2.048.ps} \\

{\small Fine $40\times40$} &
{\small Fully upscaled to $5\times5$} &
{\small Coarse $5\times5$} \\

{\tiny ~} & {\tiny ~} & {\tiny ~} \\

\multicolumn{3}{c}{\large Water saturation contours at 500 days} \\

\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{s1.124.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{s0.092.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{s2.092.ps} \\

{\small Fine $40\times40$} &
{\small Fully upscaled to $5\times5$} &
{\small Coarse $5\times5$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}

% ----------------------------------------------------------------------
% Pages 2-?: Description of the two-level scheme
% ----------------------------------------------------------------------

\newpage

%\vspace*{0.5in}
\textcolor{greyblue}{\LARGE Adaptation to a Two-Level Scheme:}

{\scriptsize
\begin{list}{}{\setlength{\itemsep}{0pt} \setlength{\leftmargin}{1.0in}}
\item[\textcolor{bo}{$A$}:] matrix which describes the interactions among cells
\item[\textcolor{bo}{$\mathbf{x}$}:] pressure unknowns
\item[\textcolor{bo}{$\mathbf{b}$}:] information from sources, boundary terms,
and gravity
\end{list}
}

{\small
Then the problem is to solve the linear system
\textcolor{bo}{$A \mathbf{x} = \mathbf{b}$}.
}

{\scriptsize
\begin{list}{}{\setlength{\itemsep}{0pt} \setlength{\leftmargin}{1.0in}}
\item[\textcolor{bo}{$\mathbf{x}_i$}:] $i$th guess at pressure unknowns
\item[\textcolor{bo}{$\mathbf{r}_i$}:] $i$th-stage residual $\mathbf{b} - A \mathbf{x}_i$
\item[\textcolor{bo}{$\mathbf{c}_i$}:] $i$th-stage smoothing corrector
\item[\textcolor{bo}{$B_{\mathrm{J}}$}:] diagonal elements of $A$ used in Jacobi
smoothing step
\item[\textcolor{bo}{$\tilde{A}$}:] matrix for the corresponding upscaling
MFEM problem
\item[\textcolor{bo}{$\mathbf{\tilde{c}}_i$}:] $i$th-stage upscaling corrector
(pressure and velocity unknowns)
% \item[\textcolor{bo}{$\mathbf{\tilde{b}}$}:] upscaling problem data
\item[\textcolor{bo}{$R$}:] restriction operator
$R: \Wh \rightarrow \WHh \times \mathbf{0}$
\item[\textcolor{bo}{$P$}:] prolongation operator
$P: \WHh \times \VHh \rightarrow \Wh$
\end{list}
}

\newpage

{\small
\begin{center}
The two-level scheme can be described schematically:
\end{center}
}

%\vspace{5in}
\begin{center}
{\setlength\unitlength{12pt}
\begin{picture}(60,30)(0,0)

\put(0,10){\framebox(12,10){\shortstack[l]{
\textcolor{bo}{\small START \strut} \\
\textcolor{greyblue}{\footnotesize $i=0$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x}_0 = \mathbf{0}$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{r}_0 = \mathbf{b} - A \mathbf{x}_0$ \strut}
}}}

\put(16,0){\framebox(12,12){\shortstack[l]{
\textcolor{bo}{\small UPSCALE \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{\tilde{c}}_i = \tilde{A}^{-1} R \mathbf{r}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x}_{i+1} = \mathbf{x}_i + P \mathbf{\tilde{c}}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $i \gets i+1$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{r}_i = \mathbf{b} - A \mathbf{x}_i$ \strut}
}}}

\put(16,18){\framebox(12,12){\shortstack[l]{
\textcolor{bo}{\small SMOOTH \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{c}_i = \alpha B_{\mathrm{J}}^{-1} \mathbf{r}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x}_{i+1} = \mathbf{x}_i + \mathbf{c}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $i \gets i+1$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{r}_i = \mathbf{b} - A \mathbf{x}_i$ \strut}
}}}

\put(32,12){\framebox(12,6){\shortstack[l]{
\textcolor{bo}{\small DONE? \strut} \\
\textcolor{greyblue}{\footnotesize Is $\| \mathbf{r}_i \|$ small? \strut}
}}}

\put(48,12){\framebox(12,6){\shortstack[l]{
\textcolor{bo}{\small FINISH \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x} \approx \mathbf{x}_i$ \strut}
}}}

%{\linethickness{2pt}
%\put(12,15){\vector(4,8){8.9442719}}
%\put(28,23){\vector(4,-6){7.2111026}}
%\put(32,13){\vector(-4,-8){8.9442719}}
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\put(44,15){\vector(1,0){4}}
%}

\put(30,9){ \textcolor{seafoamgreen}{\footnotesize NO} }
\put(44,16){ \textcolor{seafoamgreen}{\footnotesize YES} }

\end{picture}}
\end{center}
%start with a guess for the solution $\mathbf{x}_0$
%
%compute a corrector using smoothing
%$\mathbf{r}_0 = \mathbf{b} - A \mathbf{x}_0$
%$\mathbf{c}_0 = \alpha B^{-1}_{\mathrm{J}} \mathbf{r}_0$
%
%apply it
%$\mathbf{x}_{1} = \mathbf{x}_0 + \mathbf{c}_0$
%
%proceed to question with $i=1$
%
%does the residual $\mathbf{r}_i = \mathbf{b} - A \mathbf{x}_i$ still have too
%large a norm?
%
%yes:
%
%compute a corrector using upscaling
%$\mathbf{\tilde{c}}_i = \tilde{A}^{-1} (R \mathbf{r}_i)$
%$\mathbf{c}_i = P \mathbf{\tilde{c}}_i$
%
%apply it
%$\mathbf{x}^*_i = \mathbf{x}_i + \mathbf{c}_i$
%
%compute a corrector using smoothing
%$\mathbf{r}^*_i = \mathbf{b} - A \mathbf{x}^*_i$
%$\mathbf{c}^*_i = \alpha B^{-1}_{\mathrm{J}} \mathbf{r}^*_i$
%
%apply it
%$\mathbf{x}_{i+1} = \mathbf{x}^*_i + \mathbf{c}^*_i$
%
%back to question with $i \leftarrow i+1$
%
%no:
%
%stop.  $\mathbf{x}_i$ is the approximate solution

\newpage

\centerline{ \textcolor{greyblue}{\LARGE A Simple Test Case:} }

\begin{center}
\includegraphics[keepaspectratio=true,width=4.5in]{simple-perm-diagram}

{\small A Permeability Field}
\end{center}

\newpage

\begin{center}

{ \textcolor{greyblue}{\LARGE A ``Typical'' Result:} } \\
{\small Eigenstructure of Smoother and Upscaler}

\includegraphics[keepaspectratio=true,width=6in]{simplek-24-08-1e-1} \\
\begin{tabular}{c c}
\textcolor{cyan}{{\setlength\unitlength{4pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} {\tiny Coarse Smoothing Preconditioner} &
\textcolor{magenta}{{\setlength\unitlength{4pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} {\tiny Coarse Exact Preconditioner} \\
\textcolor{blue}{{\setlength\unitlength{4pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} {\tiny Fine Smoothing Preconditioner} &
\textcolor{red}{{\setlength\unitlength{4pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} {\tiny Subgrid Upscaling Preconditioner} \\
\end{tabular}

\end{center}

\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c}

\multicolumn{2}{c}{\large Vary $h$ ; Fix $H/h=3$ ; Fix $k_2=0.1$} \\

\includegraphics[keepaspectratio=true,width=3.25in]{simplek-06-02-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-12-04-1e-1} \\

{\small $6\times6$} &
{\small $12\times12$} \\

{\tiny ~} & {\tiny ~} \\

\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-48-16-1e-1} \\

{\small $24\times24$} &
{\small $48\times48$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}

\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c}

\multicolumn{2}{c}{\large Fix $h=1/24$ ; Vary $H/h$ ; Fix $k_2=0.1$} \\

\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-03-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-04-1e-1} \\

{\small $H/h=8$} & {\small $H/h=6$} \\
%{\tiny ~} & {\tiny ~} \\

\end{tabular}
\begin{tabular}{c c c}

\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-06-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-12-1e-1} \\

{\small $H/h=4$} & {\small $H/h=3$} & {\small $H/h=2$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}
%
%\newpage
%
%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c c}

\multicolumn{3}{c}{\large Fix $h=1/24$ ; Fix $H/h=3$ ; Vary $k_2$} \\

\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-0} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-2} \\

{\small $k_2=1e-0$} & {\small $k_2=1e-1$} & {\small $k_2=1e-2$} \\
{\tiny ~} & {\tiny ~} & {\tiny ~} \\

\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-3} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-4} &
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-5} \\

{\small $k_2=1e-3$} & {\small $k_2=1e-4$} & {\small $k_2=1e-5$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}

\newpage

\textcolor{greyblue}{\LARGE Some Conclusions:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}

\item This scheme will have excellent performance on smooth problems.  This
observation has been borne out in several other tests with more complicated
permeability fields.

\item This scheme will perform just like the simple test case with more
complicated heterogeneous permeability fields.  This observation, too, has been
borne out in practice.

\item Although we cannot prove it yet, this simple test case suggests:
\begin{list}{\textcolor{darkorange}{\small $\circ$}}{}
\item performance of the scheme depends only on $H/h$ and geometric factors,
\item performance is independent of $h$ and $k$,
\item and we can adjust the number of smoothing steps to make the two-level
scheme a contraction for any $H/h$
\end{list}

\end{list}
}

\newpage

% ----------------------------------------------------------------------
% Page ?: Results/Conclusions
% ----------------------------------------------------------------------

%\vspace*{0.25in}
\textcolor{greyblue}{\LARGE Preliminary Results:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}

\item convergence factor for one V-cycle is independent of the permeability and
dependent only on the relative sizes of the coarse and fine grids (like multigrid)

\item $\mathrm{O}(n)$ operations for both multigrid and this scheme (versus
$\mathrm{O}(n \log n)$ for Jacobi preconditioned conjugate gradients)

\item a bit easier to apply to general meshes than multigrid (only one coarse
mesh, not several)

\item wall-clock time speed-up of a factor of three or more over using Jacobi
preconditioned CG to solve a hybrid MFEM on the fine scale for a few test cases
(both smooth and heterogeneous permeabilities)

\item by replacing $\mathbf{\kappa}$ with an averaged value in the coarse,
upscaled equations, more accurate coarse flow information is sometimes obtained

\end{list}
}


\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c c c c c}

\multicolumn{6}{c}{\large Fix $h=1/24$ ; Vary $H/h$ ; Fix $k_2=0.1$} \\

\multicolumn{3}{c}{
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-03-1e-1}} &
\multicolumn{3}{c}{
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-04-1e-1}} \\

\multicolumn{3}{c}{\small $H/h=8$} & \multicolumn{3}{c}{\small $H/h=6$} \\
\multicolumn{6}{c}{\tiny ~} \\

\multicolumn{2}{c}{
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-06-1e-1}} &
\multicolumn{2}{c}{
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-08-1e-1}} &
\multicolumn{2}{c}{
\includegraphics[keepaspectratio=true,width=3.25in]{simplek-24-12-1e-1}} \\

\multicolumn{2}{c}{\small $H/h=4$} & \multicolumn{2}{c}{\small $H/h=3$} & \multicolumn{2}{c}{\small $H/h=2$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}



\end{document}
