Before revision:  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.
After revision:  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.