BPX-type preconditioners for 2nd and 4th order elliptic problems on the sphere

摘要

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 use C^0 linear elements on the sphere, and for the fourth order problem we use C^1 finite elements of Powell-Sabin type on the sphere. The main idea why 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.