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A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems


Authors: Thirupathi Gudi and Kamana Porwal
Journal: Math. Comp. 83 (2014), 579-602
MSC (2010): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2013-02728-7
Published electronically: June 19, 2013
MathSciNet review: 3143685
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Abstract: In this article, we derive an a posteriori error estimator for various discontinuous Galerkin (DG) methods that are proposed in (Wang, Han and Cheng, SIAM J. Numer. Anal., 48:708-733, 2010) for an elliptic obstacle problem. Using a key property of DG methods, we perform the analysis in a general framework. The error estimator we have obtained for DG methods is comparable with the estimator for the conforming Galerkin (CG) finite element method. In the analysis, we construct a non-linear smoothing function mapping DG finite element space to CG finite element space and use it as a key tool. The error estimator consists of a discrete Lagrange multiplier associated with the obstacle constraint. It is shown for non-over-penalized DG methods that the discrete Lagrange multiplier is uniformly stable on non-uniform meshes. Finally, numerical results demonstrating the performance of the error estimator are presented.


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Additional Information

Thirupathi Gudi
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
Email: gudi@math.iisc.ernet.in

Kamana Porwal
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
Email: kamana@math.iisc.ernet.in

DOI: https://doi.org/10.1090/S0025-5718-2013-02728-7
Keywords: Finite element, discontinuous Galerkin, a posteriori error estimate, obstacle problem, variational inequalities, Lagrange multiplier
Received by editor(s): November 1, 2011
Received by editor(s) in revised form: April 3, 2012, and June 18, 2012
Published electronically: June 19, 2013
Additional Notes: The first author’s work is supported by the UGC Center for Advanced Study
The second author’s work is supported in part by the UGC center for Advanced Study and in part by the Council of Scientific and Industrial Research (CSIR)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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