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An *hp*-BEM for high frequency scattering by convex polygons

## Markus Melenk (TU Wien)

Time harmonic accoustic scattering by a convex polygon is considered.
S. Chandler-Wilde and S. Langdon have recently proposed an integral
equation based method for the high frequency scattering
problem. Using a detailed regularity analysis of the solution, they were
able to design an h-version trial space that has approximation properties
that depend only logarithmically on the wave number. The key features are a)
the ability to identify the leading order (in the wave number) behavior of
the solution and b) a precise characterization of the solution behavior
near the vertices of the polygon. Since the approximation order is fixed, the
achievable convergence rate is algebraic. In this talk, we extend
their work to the hp-version of the BEM. It is shown that the solution
can be approximated at an exponential rate from the trial space; the problem
size required to achieve a given accuracy grows only logarithmically with the
wave number. In this talk, we also address the question of how to set up the stiffness matrix
with work independent of the wave number.

This is joint work with S. Langdon, University of Reading.