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A boundary element method for the solution of Helmholtz problems for a large range of complex wavenumbers

## Lehel Banjai (University of Zürich)

In this talk we present a method for the efficient solution of dense
linear systems arising in the boundary element method for the Helmholtz
equation. We consider problems that have a large range of complex
wavenumbers. The imaginary part of the wavenumber *κ* is always
nonnegative, so there is no exponential increase in the fundamental
solution (recall the fundamental solution in 3D: *e*^{iκr}/r).
For purely real wavenumbers we consider the Brakhage-Werner integral
formulation of the problem. For wavenumbers with positive imaginary parts,
we use the single layer representation. In both cases the integral
formulation is discretized by a standard Galerkin boundary element method.
The arising dense linear systems will be represented by a hierarchical
matrix format (a combination of *H* and *H*^{2} matrices).
To construct the hierarchical matrices we will use separable expansions of
the fundamental solution obtained from a diagonal multipole expansion,
interpolation, or adaptive cross approximation. We discuss in detail the
numerical instability of the multipole expansion for the range of complex
wavenumbers. We give complexity estimates of the storage and the cost of
matrix-vector products. To solve the linear systems we propose an
efficient preconditioner to be used in combination with an iterative
solver such as GMRES.

We have two applications in mind: In the first part of this talk,
presented by S. Sauter, the solution of the wave equation has been reduced
to the solution of a system of Helmholtz equations with a large range of
complex wavenumbers. The second application, is that of time harmonic
acoustic scattering, where the wavenumber is real. We present numerical
results only for the latter application.

This is joint work with W. Hackbusch and S. Sauter.