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

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 log^qN),q [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.