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: eiκr/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 H2 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.