IMA Journal of Numerical Analysis vol:26 issue:3 pages:563-583
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.