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A Posteriori Error Estimates of Discontinuous Galerkin Method for Nonmonotone Quasi-linear Elliptic Problems

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Abstract

In this paper, we propose and study the residual-based a posteriori error estimates of h-version of symmetric interior penalty discontinuous Galerkin method for solving a class of second order quasi-linear elliptic problems which are of nonmonotone type. Computable upper and lower bounds on the error measured in terms of a natural mesh-dependent energy norm and the broken H 1-seminorm, respectively, are derived. Numerical experiments are also provided to illustrate the performance of the proposed estimators.

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Acknowledgements

The authors wish to express their deepest gratitude to the anonymous referees who generously shared their insight and perspectives on the subject of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Yantai University, Shandong, 264005, P.R. China

    Chunjia Bi

  2. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Victor Ginting

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  1. Chunjia Bi
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  2. Victor Ginting
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Correspondence to Chunjia Bi.

Additional information

The research of C. Bi was partially supported by Shandong Province Natural Science Foundation (ZR2010AM004), Projects of Shandong Province Higher Educational Science and Technology Program (J10LA01, J11LA09), and the AMSS-PolyU Joint Research Institute for Engineering and Management Mathematics. The research of V. Ginting was partially supported by the grants from DOE (DE-FE0004832 and DE-SC0004982), the Center for Fundamentals of Subsurface Flow of the School of Energy Resources of the University of Wyoming (WYDEQ49811GNTG, WYDEQ49811PER), and from NSF (DMS-1016283).

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Bi, C., Ginting, V. A Posteriori Error Estimates of Discontinuous Galerkin Method for Nonmonotone Quasi-linear Elliptic Problems. J Sci Comput 55, 659–687 (2013). https://doi.org/10.1007/s10915-012-9651-2

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