The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces

Handle URI:
http://hdl.handle.net/10754/599886
Title:
The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces
Authors:
Chen, Yujia; Macdonald, Colin B.
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.
Citation:
Chen 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.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
Jan-2015
DOI:
10.1137/130929497
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
This work was supported by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorChen, Yujiaen
dc.contributor.authorMacdonald, Colin B.en
dc.date.accessioned2016-02-28T06:31:42Zen
dc.date.available2016-02-28T06:31:42Zen
dc.date.issued2015-01en
dc.identifier.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.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/130929497en
dc.identifier.urihttp://hdl.handle.net/10754/599886en
dc.description.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.en
dc.description.sponsorshipThis work was supported by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectClosest point methoden
dc.subjectGeometric multigrid methoden
dc.subjectLaplace-Beltrami operatoren
dc.subjectSurface elliptic problemen
dc.subjectSurface Poisson problemen
dc.titleThe Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfacesen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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