Hybridizable discontinuous Galerkin and mixed finite element methods for elliptic problems on surfaces

Bernardo Cockburn, Alan Demlow

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2609-2638
Number of pages30
JournalMathematics of Computation
Volume85
Issue number302
DOIs
StatePublished - 2016

Bibliographical note

Funding Information:
The first author was partially supported by NSF grant DMS-1115331. The second author was partially supported by NSF grant DMS-1318652.

Publisher Copyright:
© 2016 American Mathematical Society.

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