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


Authors: Francis J. Narcowich, Stephen T. Rowe and Joseph D. Ward
Journal: Math. Comp. 86 (2017), 197-231
MSC (2010): Primary 65M60, 65M12, 41A30, 41A55
DOI: https://doi.org/10.1090/mcom/3097
Published electronically: March 24, 2016
MathSciNet review: 3557798
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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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Additional Information

Francis J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: fnarc@math.tamu.edu

Stephen T. Rowe
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Address at time of publication: Sandia National Laboratories, Albuquerque, New Mexico 87185
Email: srowe@math.tamu.edu, srowe@sandia.gov

Joseph D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jward@math.tamu.edu

DOI: https://doi.org/10.1090/mcom/3097
Keywords: Meshless kernel method, Galerkin, PDEs on the sphere
Received by editor(s): May 19, 2014
Received by editor(s) in revised form: January 19, 2015, May 20, 2015, and July 14, 2015
Published electronically: March 24, 2016
Additional Notes: The research of the first author was supported by grant DMS-1211566 from the National Science Foundation.
The research of the second author was supported by grant DMS-1211566 from the National Science Foundation and Sandia National Laboratories
The research of the third author was supported by grant DMS-1211566 from the National Science Foundation
Article copyright: © Copyright 2016 American Mathematical Society