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{\large Andrew T. Barker \\
{\bf An additive Schwarz preconditioner for the weakly over-penalized symmetric interior penalty method}}

Department of Mathematics \\ Louisiana State University \\ Baton Rouge \\ LA 70803-4918
\\
{\tt andrewb@math.lsu.edu}\\
Susanne C. Brenner\\
Eun-Hee Park\\
Li-Yeng Sung\end{center}

We introduce and study numerically an additive Schwarz preconditioner for
the weakly over-penalized symmetric interior penalty method. The WOPSIP
method has many desirable properties for parallel computing. These
include small communication requirements at subdomain boundaries, the
ease with which meshes can be adaptively refined, and intrinsic
parallelism in the resulting matrix systems if the unknowns are ordered
properly. The additive Schwarz preconditioner is also well-suited for
parallel computing, and we show that the combination shows great promise
as a parallel algorithm. We prove optimal estimates for the condition
number of the preconditioned system under appropriate conditions, and
show numerically that the method scales
well on parallel machines and is effective for solving elliptic problems.


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