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A unifying theory of a posteriori error control for discontinuous Galerkin FEM

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Abstract

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes.

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Authors and Affiliations

  1. Department of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Carsten Carstensen

  2. Center for Computation and Technology, Johnston Hall, Louisiana State University, Baton Rouge, 70803, USA

    Thirupathi Gudi

  3. Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, UK

    Max Jensen

Authors
  1. Carsten Carstensen
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  2. Thirupathi Gudi
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  3. Max Jensen
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Corresponding author

Correspondence to Carsten Carstensen.

Additional information

C. Carstensen and M. Jensen supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the Hausdorff Institute of Mathematics in Bonn, Germany.

C. Carstensen, T. Gudi, and M. Jensen supported by DST-DAAD (PPP-05) project no. 32307481.

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Carstensen, C., Gudi, T. & Jensen, M. A unifying theory of a posteriori error control for discontinuous Galerkin FEM. Numer. Math. 112, 363–379 (2009). https://doi.org/10.1007/s00211-009-0223-9

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  • DOI: https://doi.org/10.1007/s00211-009-0223-9

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