Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number κ in two dimensions. We consider the Brakhage-Werner integral formulation of the problem, discretised by the Galerkin boundary element method. The dense n×n Galerkin matrix arising from this approach is represented by a sum of an H-matrix and an H2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the H2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an H-matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the H-matrix. Further, an approximate LU -decomposition of such a recompressed H-matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in two dimension are presented, where the linear systems are solved by a preconditioned iterative method.