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

Gudi, Thirupathi; Porwal, Kamana

doi:10.1090/S0025-5718-2013-02728-7

A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems
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by Thirupathi Gudi and Kamana Porwal PDF

Math. Comp. 83 (2014), 579-602 Request permission

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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