A novel Galerkin method for solving PDES on the sphere using highly localized kernel bases

Narcowich, Francis; Rowe, Stephen; Ward, Joseph

doi:10.1090/mcom/3097

A novel Galerkin method for solving PDES on the sphere using highly localized kernel bases
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by Francis J. Narcowich, Stephen T. Rowe and Joseph D. Ward PDF

Math. Comp. 86 (2017), 197-231 Request permission

Abstract:

The main goal of this paper is to introduce a novel meshless kernel Galerkin method for numerically solving partial differential equations on the sphere. Specifically, we will use this method to treat the partial differential equation for stationary heat conduction on $\mathbb S^2$, in an inhomogeneous, anisotropic medium. The Galerkin method used to do this employs spatially well-localized, “small footprint”, robust bases for the associated kernel space. The stiffness matrices arising in the problem have entries decaying exponentially fast away from the diagonal. Discretization is achieved by first zeroing out small entries, resulting in a sparse matrix, and then replacing the remaining entries by ones computed via a very efficient kernel quadrature formula for the sphere. Error estimates for the approximate Galerkin solution are also obtained.

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