Solving the 3D Laplace equation by meshless collocation via harmonic kernels
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.
scientific article; zbMATH DE number 6153876