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Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations


Authors: Sangita Yadav, Amiya K. Pani and Eun-Jae Park
Journal: Math. Comp. 82 (2013), 1297-1335
MSC (2010): Primary 65-XX
DOI: https://doi.org/10.1090/S0025-5718-2013-02662-2
Published electronically: January 24, 2013
MathSciNet review: 3042565
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Abstract: Based on the analysis of Cockburn et al. [Math. Comp. 78 (2009), pp. 1-24] for a selfadjoint linear elliptic equation, we first discuss superconvergence results for nonselfadjoint linear elliptic problems using discontinuous Galerkin methods. Further, we have extended our analysis to derive superconvergence results for quasilinear elliptic problems. When piecewise polynomials of degree $ k\geq 1$ are used to approximate both the potential as well as the flux, it is shown, in this article, that the error estimate for the discrete flux in $ L^2$-norm is of order $ k+1.$ Further, based on solving a discrete linear elliptic problem at each element, a suitable postprocessing of the discrete potential is developed and then, it is proved that the resulting post-processed potential converges with order of convergence $ k+2 $ in $ L^2$-norm. These results confirm superconvergent results for linear elliptic problems.


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Additional Information

Sangita Yadav
Affiliation: Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076
Email: sangita@.iitk@gmail.com

Amiya K. Pani
Affiliation: Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076
Email: akp@math.iitb.ac.in

Eun-Jae Park
Affiliation: Department of Computational Science and Engineering-WCU, Yonsei University
Email: ejpark@yonsei.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-2013-02662-2
Received by editor(s): October 29, 2010
Received by editor(s) in revised form: November 15, 2011
Published electronically: January 24, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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