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

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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 theH 1/2-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 orderO(N logq N),q ε [2, 3], with memory needsO(N log2 N), whereN 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.

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia

    Boris N. Khoromskij

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117, Berlin, Germany

    Siegfried Prössdorf

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  1. Boris N. Khoromskij
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  2. Siegfried Prössdorf
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Communicated by C.A. Micchelli

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Khoromskij, B.N., Prössdorf, S. Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation. Adv Comput Math 4, 331–355 (1995). https://doi.org/10.1007/BF02123480

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  • DOI: https://doi.org/10.1007/BF02123480

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