  This work develops optimal preconditioners for the discrete H(curl) and
H(div) problems on two-dimensional surfaces by nodal auxiliary space
preconditioning [R. Hiptmair, J. Xu: SIAM J. Numer. Anal. \textbf{45},
2483-2509 (2007)]. In particular, on unstructured triangulated surfaces, we
develop fast and user-friendly preconditioners for the edge and face element
discretizations of curl-curl and grad-div problems based on inverting several
discrete surface Laplacians. The proposed preconditioners lead to efficient
iterative methods for computing harmonic tangential vector fields on discrete
surfaces. Numerical experiments on two- and three-dimensional hypersurfaces are
presented to test the performance of those surface preconditioners.
