Abstract
We present a meshless technique which can be seen as an alternative to the method of fundamental solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artificial boundary needed, and there is a fairly general and complete error analysis using standard techniques from meshless methods for the recovery of functions. We present two explicit examples of harmonic kernels, a mathematical analysis providing error bounds and convergence rates, and some illustrating numerical examples.
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Communicated by Yuesheng Xu.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Schaback, R. Solving the Laplace equation by meshless collocation using harmonic kernels. Adv Comput Math 31, 457 (2009). https://doi.org/10.1007/s10444-008-9078-3
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DOI: https://doi.org/10.1007/s10444-008-9078-3