WIAS Preprint No. 150, (1995)

Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation



Authors

  • Khoromskij, Boris N.
  • Prössdorf, Siegfried

2010 Mathematics Subject Classification

  • 65N20 65N30 65P10

Keywords

  • Boundary integral equations, domain decomposition, fast elliptic problem solvers, interface operators, matrix compression, multilevel preconditioning

DOI

10.20347/WIAS.PREPRINT.150

Abstract

In this paper we propose and analyze some strategies to construct asymptotically optimal algorithms for solving boundary reductions of the Laplace equation in the interior and exterior of a polygon. The interior Dirichlet or Neumann problems are, in fact equivalent to a direct treatment of the Dirichlet-Neumann mapping or its inverse i.e. the Poincaré-Steklov (PS) operator. To construct a fast algorithm for the treatment of the discrete PS operator in the case of polygons composed of rectangles and regular right triangles, we apply the Bramble-Pasciak-Xu (BPX) multilevel preconditioner to the equivalent interface problem in the H1/2-setting. Furthermore, a fast matrix-vector multiplication algorithm is based on the frequency cutting techniques applied to the local Schur complements associated with the rectangular substructures specifying the nonmatching decomposition of a given polygon. The proposed compression scheme to compute the action of the discrete interior PS operator is shown to have a complexity of the order O(N logq N), q ∈ [2,3] with memory needs of O(N log2 N) where N is the number of degrees of freedom on the polygonal boundary under consideration. In the case of exterior problems we propose a modification of the standard direct BEM whose implementation is reduced to the wavelet approximation applied to either single layer or hypersingular harmonic potentials and, in addition, to the matrix-vector multiplication for the discrete interior PS operator.

Appeared in

  • Advances in Computational Mathematics 4 (1995) pp. 331--355

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