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A Posteriori Error Control for a Weakly Over-Penalized Symmetric Interior Penalty Method

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Abstract

A reliable and efficient residual based a posteriori error estimator is constructed for a weakly over-penalized symmetric interior penalty method for second order elliptic problems. Numerical results that demonstrate the performance of the error estimator are presented.

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References

  1. Ainsworth, M.: A synthesis of a posteriori error estimation techniques for conforming, non-conforming and discontinuous Galerkin finite element methods. In: Recent Advances in Adaptive Computation. Contemp. Math., vol. 383, pp. 1–14. Am. Math. Soc., Providence (2005)

    Google Scholar 

  2. Ainsworth, M.: A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45, 1777–1798 (2007) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. Wiley-Interscience, New York (2000)

    MATH  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001/02)

    Article  MathSciNet  Google Scholar 

  5. Becker, R., Hansbo, P., Larson, M.G.: Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 192, 723–733 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenner, S.C., Li, F., Sung, L.-Y.: A locally divergence-free interior penalty method for two dimensional curl-curl problems. SIAM J. Numer. Anal. 46, 1190–1211 (2008)

    Article  MathSciNet  Google Scholar 

  7. Brenner, S.C., Owens, L., Sung, L.-Y.: A weakly over-penalized symmetric interior penalty method. Electron. Trans. Numer. Anal. 30, 107–127 (2008)

    MathSciNet  Google Scholar 

  8. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    MATH  Google Scholar 

  9. Bustinza, R., Gatica, G.N., Cockburn, B.: An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput. 22/23, 147–185 (2005)

    Article  MathSciNet  Google Scholar 

  10. Carstensen, C., Gudi, T., Jensen, M.: A unifying theory of a posteriori error control for discontinuous Galerkin fem. Preprint (2007). Numer. Math. (to appear)

  11. Castillo, P.: An a posteriori error estimate for the local discontinuous Galerkin method. J. Sci. Comput. 22/23, 187–204 (2005)

    Article  MathSciNet  Google Scholar 

  12. Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7, 33–75 (1973)

    MathSciNet  Google Scholar 

  13. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Houston, P., Schötzau, D., Wihler, T.P.: Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci. 17, 33–62 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karakashian, O.A., Pascal, F.: Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45, 641–665 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Owens, L.: Multigrid methods for weakly over-penalized interior penalty methods. Ph.D. thesis, University of South Carolina (2007)

  18. Rivière, B., Wheeler, M.F.: A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46, 141–163 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schneider, R., Xu, Y., Zhou, A.: An analysis of discontinuous Galerkin methods for elliptic problems. Adv. Comput. Math. 25, 259–286 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. In: Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), vol. 50, pp. 67–83 (1994)

  21. Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1995)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    Susanne C. Brenner

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

    Thirupathi Gudi

  3. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Li-yeng Sung

Authors
  1. Susanne C. Brenner
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  2. Thirupathi Gudi
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  3. Li-yeng Sung
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Corresponding author

Correspondence to Susanne C. Brenner.

Additional information

The work of S.C. Brenner was supported in part by the National Science Foundation under Grant No. DMS-07-38028, and Grant No. DMS-07-13835.

The work of L. Sung was supported in part by the National Science Foundation under Grant No. DMS-07-13835.

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Brenner, S.C., Gudi, T. & Sung, Ly. A Posteriori Error Control for a Weakly Over-Penalized Symmetric Interior Penalty Method. J Sci Comput 40, 37–50 (2009). https://doi.org/10.1007/s10915-009-9278-0

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  • DOI: https://doi.org/10.1007/s10915-009-9278-0

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