The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces
KAUST Grant NumberKUK-C1-013-04
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Abstract© 2015 Society for Industrial and Applied Mathematics. Elliptic partial differential equations are important from both application and analysis points of view. In this paper we apply the closest point method to solve elliptic equations on general curved surfaces. Based on the closest point representation of the underlying surface, we formulate an embedding equation for the surface elliptic problem, then discretize it using standard finite differences and interpolation schemes on banded but uniform Cartesian grids. We prove the convergence of the difference scheme for the Poisson's equation on a smooth closed curve. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method in the setting of the closest point method. Convergence studies in both the accuracy of the difference scheme and the speed of the multigrid algorithm show that our approaches are effective.
CitationChen Y, Macdonald CB (2015) The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces. SIAM Journal on Scientific Computing 37: A134–A155. Available: http://dx.doi.org/10.1137/130929497.
SponsorsThis work was supported by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).