Preconditioners for pseudodifferential equations on the sphere with radial basis functions

Authors:
T. Tran;Q. T. Le Gia;I. H. Sloan;E. P. Stephan
Affiliations:
The University of New South Wales, School of Mathematics and Statistics, 2052, Sydney, Australia;The University of New South Wales, School of Mathematics and Statistics, 2052, Sydney, Australia;The University of New South Wales, School of Mathematics and Statistics, 2052, Sydney, Australia;Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167, Hannover, Germany
Venue:
Numerische Mathematik
Year:
2010

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Abstract

In a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the Galerkin approximation of elliptic partial differential equations on the sphere. In this paper the methods are extended to more general pseudodifferential equations on the sphere, using as before spherical radial basis functions for the approximation space, and again preconditioning the ill-conditioned linear systems of the Galerkin approximation by the additive Schwarz method. Numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere $${\mathbb{S}^2}$$.