Fast auxiliary space preconditioners on surfaces

This work presents new uniform preconditioners for the discrete Laplace-Beltrami operator on hypersurfaces. In particular, within the framework of fast auxiliary space preconditioning (FASP), we develop optimal multilevel preconditioners for the Laplace-Beltrami type equation discretized by surface Lagrange, nonconforming linear, and discontinuous Galerkin elements. The framework naturally deals with semi-definite problems on a closed surface. Numerical experiments are presented to illustrate the efficiency of our preconditioners.