Abstract
In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 2-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 Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -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 2D are presented, where the linear systems are solved by a preconditioned iterative method.
Banjai, L