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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator


Authors: Andrea Bonito and Joseph E. Pasciak
Journal: Math. Comp. 81 (2012), 1263-1288
MSC (2010): Primary 65N30, 65N55
Published electronically: October 13, 2011
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Abstract: We design and analyze variational and non-variational multigrid algorithms for the Laplace-Beltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the $ V$-cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost.

Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio.


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Additional Information

Andrea Bonito
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: bonito@math.tamu.edu

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: pasciak@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02551-2
PII: S 0025-5718(2011)02551-2
Received by editor(s): August 23, 2009
Received by editor(s) in revised form: March 23, 2011
Published electronically: October 13, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.