A PETROV-GALERKIN KERNEL APPROXIMATION ON THE SPHERE

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov-Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appro...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2018-01, Vol.56 (1), p.274-295
Main Author: MIRZAEI, DAVOUD
Format: Article
Language:English
Subjects:
condition numbers
convo-lution data
error analysis
Petrov–Galerkin method
radial basis functions
spherical basis functions
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Summary:In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov-Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration.
ISSN:0036-1429
1095-7170
1095-7170
DOI:10.1137/16M1106626