Hierarchical matrix techniques for low- and high-frequency Helmholtz problems
Banjai, Lehel ; Hackbusch, Wolfgang
In: IMA Journal of Numerical Analysis, 2007, vol. 28, no. 1, p. 46-79
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- In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number κ in 2D. We consider the Brakhage-Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n × n Galerkin matrix arising from this approach is represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the 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 -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the -matrix. Further, an approximate LU decomposition of such a recompressed -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