Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Abstract

This paper solves the Laplace equation Δu?=?0 on domains ?????3 by meshless collocation on scattered points of the boundary $\partial\Omega$ . Due to the use of new positive definite kernels K(x, y) which are harmonic in both arguments and have no singularities for x?=?y, one can directly interpolate on the boundary, and there is no artificial boundary needed as in the Method of Fundamental Solutions. In contrast to many other techniques, e.g. the Boundary Point Method or the Method of Fundamental Solutions, we provide a solid and comprehensive mathematical foundation which includes error bounds and works for general star-shaped domains. The convergence rates depend only on the smoothness of the domain and the boundary data. Some numerical examples are included.