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

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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 Poincare-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 theH1/2-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 logqN),q e [2, 3], with memory needsO(N log2N), 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.