Nodal Auxiliary Space Preconditioning for the Surface de Rham Complex

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