A hierarchical basis preconditioner for the biharmonic equation on the sphere

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 derive 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 accounts for 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