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 R3. 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 C1 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.