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A numerical study of some radial basis function based solution methods for elliptic PDEs. (English) Zbl 1049.65136
Summary: During the last decade, three main variations have been proposed for solving elliptic partial differential equations (PDEs) by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ill conditioning, which has previously limited the range that could be used for this parameter.
We find that the best values for it often fall outside the range that was previously available. We have also looked at piecewise smooth versus infinitely smooth RBFs, and found that for PDE applications with smooth solutions, the infinitely smooth RBFs are preferable, mainly because they lead to higher accuracy. In a comparison of RBF-based methods against two standard techniques (a second-order finite difference method and a pseudospectral method), the former gave a much superior accuracy.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N06 Finite difference methods for boundary value problems involving PDEs
Software:
Matlab
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