===affil2:

===firstname:
Bram
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===firstname3:

===lastname2:

===lastname:
Metsch
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===lastname3:

===email:
metsch@ins.uni-bonn.de
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===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===affil7:

===firstname7:

===postal:
Universitaet Bonn
Institut fuer Numerische Simulation
Wegelerstrasse 6
53115 Bonn
GERMANY
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===ABSTRACT:
We present an Algebraic Multigrid (AMG) approach to the solution of Stokes-type saddle point problems of the form
\begin{equation*}
  \mathcal{K}: \begin{pmatrix} \mathcal{V} \\ \mathcal{W} \end{pmatrix} \rightarrow\begin{pmatrix} \mathcal{V} \\ \mathcal{W} \end{pmatrix}, \qquad  \mathcal{K} = \begin{pmatrix} A & B^T \\ B & -C \end{pmatrix}
\end{equation*}
where $A$ is positive definite and $C$ is positive semi-definite.\\
%
In particular, we address the two main challenges to the
construction of a saddle point AMG method,
\begin{enumerate}
\item The indefinite matrix $\mathcal{K}$ does not define an inner product
  and norm, hence, it is not possible to carry out the AMG convergence
  proof in terms of the energy norm (as is done in AMG for symmetric
  positive matrices),
\item The coarse grid matrix computed by the Galerkin product may not
  be invertible at all.
\end{enumerate}
The starting point for the construction of our AMG method for saddle
point systems is the inexact symmetric Uzawa smoothing scheme
introduced in \cite{Schoeberl.Zulehner:2003}. We set up the
coarse grids and prolongation operators for the spaces $\mathcal{V}$
and $\mathcal{W}$ independently. From these per-space interpolation operators,
we assemble the global interpolation and restriction operators $\mathcal{P}$ and $\mathcal{R}$ such that an inf-sup condition for $K$ implies an inf-sup-condition for the coarse system computed by the Galerkin product
\begin{equation*}
\mathcal{K}^C = \mathcal{R} \mathcal{K} \mathcal{P}.
\end{equation*}
Especially, our approach does not require that the coarse grids for
$\mathcal{V}$ and $\mathcal{W}$ are adapted to each other. Instead,
the stability and thus the invertibility of the coarse grid operator is ensured by the
construction of the transfer operators.\\
We also give a two-grid convergence proof for our method based
on the framework of \cite{Notay:2009}, which does not depend on the
existence of an energy norm.
%
\bibliographystyle{plain}
\begin{thebibliography}{10}

\bibitem{Schoeberl.Zulehner:2003}
{\sc J. Sch\"oberl and W. Zulehner}. {On Schwarz-type Smoothers for Saddle Point Problems}. Numerische Mathematik 95 (2003) 377-399.

\bibitem{Notay:2009}
{\sc Y. Notay}. {Algebraic analysis of two-grid methods: The nonsymmetric case}. Numer. Linear Algebra Appl. 17 (2009) 73-96.

\end{thebibliography}
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===lastname5:

===affilother:

===title:
Algebraic Multigrid (AMG) for Saddle Point Systems - A self-stabilizing approach
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