A wavelet algorithm for the boundary element solution of a geodetic boundary value problem

Abstract

In this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a part of the surface of the earth. This singular equation is the boundary integral equation corresponding to the oblique derivative boundary problem for Laplace's equation. We introduce special wavelet bases for the spaces of test and trial functions. Analogously to well-known results on wavelet algorithms, the stiffness matrices with respect to these bases can be reduced to sparse matrices such that the assembling of the matrices and the iterative solution of the matrix equations quicken. Though the theoretical results apply only to integral equations with ‘smooth’ solutions over ‘smooth’ manifolds, we present numerical tests for a geometry as difficult as the surface of the earth.