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

This paper solves the Laplace equation ∆u = 0 on domains Ω ⊂ R by meshless collocation on scattered points of the boundary ∂Ω. In contrast to the Method of Fundamental Solutions, there are no singularities and no artificial boundaries, since we use new singularity–free positive definite kernels which are harmonic in both arguments. In contrast to many other techniques, e.g. the Boundary Point Method or the Method of Fundamental Solutions, we provide a solid 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.