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


Authors: Bernardo Cockburn and Alan Demlow
Journal: Math. Comp. 85 (2016), 2609-2638
MSC (2010): Primary 58J32, 65N15, 65N30
DOI: https://doi.org/10.1090/mcom/3093
Published electronically: March 4, 2016
MathSciNet review: 3522964
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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.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minnesota 55455 – and – Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dahran, Saudi Arabia
Email: cockburn@math.umn.edu

Alan Demlow
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368
Email: demlow@math.tamu.edu

DOI: https://doi.org/10.1090/mcom/3093
Keywords: Laplace-Beltrami operator, surface finite element methods, a priori error estimates, boundary value problems on surfaces, discontinuous Galerkin methods, hybridizable finite element methods, mixed finite element methods
Received by editor(s): June 19, 2014
Received by editor(s) in revised form: February 17, 2015
Published electronically: March 4, 2016
Additional Notes: The first author was partially supported by NSF grant DMS-1115331.
The second author was partially supported by NSF grant DMS-1318652.
Article copyright: © Copyright 2016 American Mathematical Society

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