BPX‐type Preconditioners for Second and Fourth Order Elliptic Problems on the Sphere

Abstract

We develop two Bramble–Pasciak–Xu‐type preconditioners for second (resp., fourth) order elliptic problems on the surface of the two‐sphere. To discretize the second order problem we construct $C^0$ linear elements on the sphere, and for the fourth order problem we construct $C^1$ finite elements of Powell–Sabin type on the sphere. The main reason these BPX preconditioners work depends on this particular choice of basis. We prove optimality and provide numerical examples. Furthermore we numerically compare the BPX preconditioners with the suboptimal hierarchical basis preconditioners.