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

Authors:
Y. C. Hon;R. Schaback
Affiliations:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR;Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, Germany 37083
Venue:
Advances in Computational Mathematics
Year:
2013

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Abstract

This paper solves the Laplace equation Δu驴=驴0 on domains 驴驴驴驴驴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驴=驴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.