Title: A hierarchical basis preconditioner for the biharmonic equation on the sphere
Authors: Maes, Jan ×
Bultheel, Adhemar #
Issue Date: Jul-2006
Publisher: Academic Press
Series Title: IMA Journal of Numerical Analysis vol:26 issue:3 pages:563-583
Abstract: In this paper, we propose a natural way to extend a bivariate Powell-Sabin (PS) B-spline basis on a planar polygonal domain to a PS B-spline basis defined on a subset of the unit sphere in R^3. The spherical basis inherits many properties of the bivariate basis such as local support, the partition of unity property and stability. This allows us to construct a C^1 continuous hierarchical basis on the sphere that is suitable for preconditioning fourth-order elliptic problems on the sphere. We show that the stiffness matrix relative to this hierarchical basis has a logarithmically growing condition number, which is a suboptimal result compared to standard multigrid methods. Nevertheless, this is a huge improvement over solving the discretized system without preconditioning, and its extreme simplicity contributes to its attractiveness. Furthermore, we briefly describe a way to stabilize the hierarchical basis with the aid of the lifting scheme. This yields a wavelet basis on the sphere for which we find a uniformly well-conditioned and (quasi-) sparse stiffness matrix.
ISSN: 0272-4979
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Numerical Analysis and Applied Mathematics Section
× corresponding author
# (joint) last author

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