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

Type
Article

Authors
Bonito, Andrea
Pasciak, Joseph E.

KAUST Grant Number
KUS-C1-016-04

Date
2012-09-01

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. © 2011 American Mathematical Society.

Citation
Bonito A, Pasciak JE (2012) Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator. Math Comp 81: 1263–1288. Available: http://dx.doi.org/10.1090/s0025-5718-2011-02551-2.

Acknowledgements
This work was supported in part by award number KUS-C1-016-04 made by KingAbdulla University of Science and Technology (KAUST). The first author was alsosupported in part by the National Science Foundation through Grant DMS-0914977while the second was also supported in part by the National Science Foundationthrough Grant DMS-0609544.

Publisher
American Mathematical Society (AMS)

Journal
Mathematics of Computation

DOI
10.1090/s0025-5718-2011-02551-2

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