Solving the 3D Laplace equation by meshless collocation via harmonic kernels
type: Item
instance of: scholarly article
title: Solving the 3D Laplace equation by meshless collocation via harmonic kernels
zbMATH Open document ID: 1272.65096
author: Robert Schaback
author: Y. C. Hon
published in: Advances in Computational Mathematics
publication date: 2013-04-15T00:00:00Z
review text: The authors consider a meshless collocation method for the numerical solution of the Laplace equation in a star-like domain in 3D along with Dirichlet boundary data. They consider the harmonic extension too, but to avoid the summations needed they propose two kernel functions delivering the approximate solution. Proving their positive definiteness they are able to invoke results of \textit{F. J. Narcowich} and \textit{J. D. Ward} on scattered data interpolatoin on spheres [SIAM J. Math. Anal. 33, No. 6, 1393-1410 (2002; Zbl 1055.41007)] and to conclude that the solution error is dominated by the interpolation error on the boundary and to give an error bound in \(h\) depending on the smoothness of the boundary data. Their numerical experiments show that best results are obtained for harmonic boundary data whereas otherwise convergence may be slow. In any case, good algorithms for the choice of scattered points on the boundary are needed.
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 65N80
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 65N15
zbMATH DE Number: 6153876
zbMATH Keywords: 3D Laplace equation
zbMATH Keywords: pointless collocation method
zbMATH Keywords: harmonic extension
zbMATH Keywords: positive definite kernels
zbMATH Keywords: error bounds
zbMATH Keywords: meshless collocation method
zbMATH Keywords: numerical experiments
zbMATH Keywords: convergence
zbMATH Keywords: algorithm
describes a project that uses: SumTools
MaRDI profile type: Publication
full work available at URL: s10444-011-9224-1
OpenAlex ID: W2055962168
cites work: Q5558293
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cites work: Solving the Laplace equation by meshless collocation using harmonic kernels
DOI: 10.1007/S10444-011-9224-1
description: scientific article; zbMATH DE number 6153876