We define and analyze hybridizable discontinuous Galerkin methods for the Laplace-Beltrami problem on implicitly defined surfaces. We show that the methods can retain the same convergence and superconvergence properties they enjoy in the case of flat surfaces. Special attention is paid to the relative effect of approximation of the surface and that introduced by discretizing the equations. In particular, we show that when the geometry is approximated by polynomials of the same degree of those used to approximate the solution, although the optimality of the approximations is preserved, the superconvergence is lost. To recover it, the surface has to be approximated by polynomials of one additional degree. We also consider mixed surface finite element methods as a natural part of our presentation. Numerical experiments verifying and complementing our theoretical results are shown.
Bibliographical noteFunding Information:
The first author was partially supported by NSF grant DMS-1115331. The second author was partially supported by NSF grant DMS-1318652.
© 2016 American Mathematical Society.