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A Posteriori Local Discontinuous Galerkin Error Estimation for Two-Dimensional Convection–Diffusion Problems

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Abstract

We present a simple, efficient, and asymptotically correct a posteriori error estimates for a minimal dissipation local discontinuous Galerkin method applied to two-dimensional diffusion and convection–diffusion problems on rectangular meshes. The finite element spaces are obtained by performing a local error analysis and a posteriori error estimates are computed by solving local problems on each element. We present computational results for several problems to show the efficiency and accuracy of our error estimates. It is shown that even in the presence of boundary layers our error estimates converge to the true error under mesh refinement when Shishkin meshes are used.

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Acknowledgments

This research was partially supported by the National Science Foundation (Grant Number DMS 0809262), the NASA Nebraska Space Grant Program at the University of Nebraska at Omaha, and the University Committee on Research and Creative Activity (UCRCA) at the University of Nebraska at Omaha. The authors would also like to thank the referees for their constructive comments and remarks which helped improve the quality and readability of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Nebraska at Omaha, Omaha, NE , 68182, USA

    Mahboub Baccouch

  2. Department of Mathematics, Virginia Tech, Blacksburg, VA , 24061, USA

    Slimane Adjerid

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  1. Mahboub Baccouch
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  2. Slimane Adjerid
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Correspondence to Mahboub Baccouch.

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Baccouch, M., Adjerid, S. A Posteriori Local Discontinuous Galerkin Error Estimation for Two-Dimensional Convection–Diffusion Problems. J Sci Comput 62, 399–430 (2015). https://doi.org/10.1007/s10915-014-9861-x

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  • DOI: https://doi.org/10.1007/s10915-014-9861-x

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