An overlapping additive Schwarz preconditioner for the Laplace-Beltrami equation using spherical splines

Authors:
Duong Pham;Thanh Tran;Simon Crothers
Affiliations:
School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia 2052;School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia 2052;School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia 2052
Venue:
Advances in Computational Mathematics
Year:
2012

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Abstract

We present an overlapping domain decomposition technique for solving the Laplace---Beltrami equation on the sphere with spherical splines. We prove that the condition number of the additive Schwarz operator is bounded by $O\left(H^2/h^2\right)$ , where H and h are the sizes of the coarse and fine meshes, respectively. In the case that the degree of the splines is even, a better bound $O\left(\max_{1\leq k \leq J}\left(1+H_k/\delta_k\right)\right)$ is proved. Here J is the number of subdomains, H k is the size of the kth subdomain, and 驴 k is the size of the overlap of the kth subdomain. The method is illustrated by numerical experiments on large point sets taken from magsat satellite data.