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An analysis of discontinuous Galerkin methods for elliptic problems

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Abstract

We study global and local behaviors for three kinds of discontinuous Galerkin schemes for elliptic equations of second order. We particularly investigate several a posteriori error estimations for the discontinuous Galerkin schemes. These theoretical results are applied to develop local/parallel and adaptive finite element methods, based on the discontinuous Galerkin methods.

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References

  1. R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975).

    MATH  Google Scholar 

  2. D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982) 742–760.

    Article  MATH  MathSciNet  Google Scholar 

  3. D.N. Arnold, F. Brezzi, B. Cockburn and D. Marini, Discontinuous Galerkin methods for elliptic problems, in: Discontinuous Galerkin Methods, eds. B. Cockburn, G.E. Karniadakis and C.-W. Shu, Lectures in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 89–101.

    Google Scholar 

  4. D.N. Arnold, F. Brezzi, B. Cockburn and D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/2002) 1749–1779.

    Article  MathSciNet  Google Scholar 

  5. I. Babuska, The finite element method with penalty, Math. Comp. 27 (1973) 221–228.

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Babuska and A.D. Miller, A feedback finite element method with a posteriori error estimation, Part 1, Comput. Methods Appl. Mech. Engrg. 61 (1987) 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  7. I. Babuska and C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736–754.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Babuska, O.C. Zienkiewicz, J. Gago and E.R. de A. Oliveira, Accuracy Estimates and Adaptive Refinements in Finite Element Computations (Wiley, New York, 1986).

    MATH  Google Scholar 

  9. I. Babuska and M. Zlamal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973) 863–875.

    Article  MathSciNet  MATH  Google Scholar 

  10. R.E. Bank, Hierarchical bases and the finite element method, Acta Numerica 5 (1996) 1–43.

    Article  MATH  MathSciNet  Google Scholar 

  11. R.E. Bank and M. Holst, A new paradigm for parallel adaptive meshing algorithms, SIAM J. Sci. Comput. 22 (2001) 1411–1443.

    Article  MathSciNet  Google Scholar 

  12. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, J. Comput. Phys. 131 (1997) 267–279.

    Article  MathSciNet  MATH  Google Scholar 

  13. C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection–diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999) 311–341.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Becker and P. Hansbo, A finite element method for domain decomposition with non-matching grids, Report No. 3616, RINRIA (1999).

  15. C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998) 1893–1916.

    Article  MathSciNet  MATH  Google Scholar 

  16. C.L. Bottasso, S. Micheletti and R. Sacco, The discontinuous Petrov–Galerkin method for elliptic problems, Comput. Methods Appl. Mech. Engrg. 191 (2002) 3391–3409.

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations 16 (2000) 365–378.

    Article  MathSciNet  MATH  Google Scholar 

  18. C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for linear finite element methods, SIAM J. Numer. Anal. 36 (1999) 1571–1587.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000) 1676–1706.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Castillo, B. Cockburn, D. Schötzau and C. Schwab, Optimal a priori error estimates for thehp-version of the local discontinuous Galerkin method for convection–diffusion problems, Math. Comp. 71 (2002), 455–478.

    Article  MathSciNet  MATH  Google Scholar 

  21. P.G. Ciarlet and J.L. Lions, Handbook of Numerical Analysis, Vol. II, Finite Element Methods (Part I) (North-Holland, Amsterdam, 1991).

    MATH  Google Scholar 

  22. P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numer. 9 (1975) 77–84.

    MATH  Google Scholar 

  23. B. Cockburn, G.E. Karniadakis and C.-W. Shu, Discontinuous Galerkin Methods, Lectures in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000).

    MATH  Google Scholar 

  24. B. Cockburn, G.E. Karniadakis and C.-W. Shu, The development of discontinuous Galerkin methods, in: Discontinuous Galerkin Methods, eds. B. Cockburn, G.E. Karniadakis and C.-W. Shu, Lectures in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 1–50.

    Google Scholar 

  25. J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in: Lecture Notes in Physics, Vol. 58 (1976) pp. 207–216.

  26. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica (1995) 105–158.

  27. Q. Lin and A. Zhou, Some arguments for recovering the finite element error of hyperbolic problems, Acta Math. Sci. 11 (1991) 471–476.

    MathSciNet  Google Scholar 

  28. J.A. Nitsche, Über ein Variationsprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unteworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15.

    Article  MATH  MathSciNet  Google Scholar 

  29. J.A. Nitsche and A.H. Schatz, Interiori estimates for Ritz–Galerkin methods, Math. Comp. 28 (1974) 937–955.

    Article  MathSciNet  MATH  Google Scholar 

  30. J.T. Oden, I. Babuska and C.E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys. 146 (1998) 491–519.

    Article  MathSciNet  MATH  Google Scholar 

  31. B. Riviere and F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations, in: Discontinuous Galerkin Methods, eds. B. Cockburn, G.E. Karniadakis and C.-W. Shu, Lectures in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 231–244.

    Google Scholar 

  32. B. Riviere, M.F. Wheeler and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (2001) 902–931.

    Article  MathSciNet  MATH  Google Scholar 

  33. A.H. Schatz and L.B. Wahlbin, Interior maximum-norm estimates for finite element methods, Math. Comp. 31 (1977) 414–442.

    Article  MathSciNet  MATH  Google Scholar 

  34. A.H. Schatz and L.B. Wahlbin, Interior maximum-norm estimates for finite element methods, Part II, Math. Comp. 64 (1995) 907–928.

    Article  MathSciNet  MATH  Google Scholar 

  35. L.R. Scott and S. Zhang, Finite element interpolation of mpon-smooth functions satisfying boundary conditions, Math. Comp. 54 (1992) 483–493.

    Article  MathSciNet  Google Scholar 

  36. E. Süli, C. Schwab and P. Houston, hp-DGFEM for partial differential equations with nonnegative characteristic form, in: Discontinuous Galerkin Methods, eds. B. Cockburn, G.E. Karniadakis and C.-W. Shu, Lectures in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 221–230.

    Google Scholar 

  37. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62 (1994) 445–475.

    Article  MATH  MathSciNet  Google Scholar 

  38. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations, Bericht Nr. 180, Fakultät für Mathematik, Ruhr-Universität Bochum (1995).

  39. R. Verfürth, A Review of A-Posteriori Error Estimation and Adaptive Mesh Refinement (Wiley/Teubner, 1996).

  40. R. Verfürth, A posteriori error estimators for convection-diffusion equations, Numer. Math. 80 (1998) 641–663.

    Article  MATH  MathSciNet  Google Scholar 

  41. L.B. Wahlbin, Local behavior in finite element methods, in [17] (1991) 355–522.

  42. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978) 152–161.

    Article  MATH  MathSciNet  Google Scholar 

  43. B.I. Wohlmuth, A residual based error estimator for mortar finite element discretizations, Numer. Math. 84 (1999) 143–171.

    Article  MATH  MathSciNet  Google Scholar 

  44. J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000) 881–909.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Xu and A. Zhou, Locality and parallelization of finite elements, in: Proc. of the 11th Domain Decomposition Methods, eds. C.H. Lai, P.E. Bjørstad, M. Cross and O.B. Widlund, DDM.org (1999) pp. 140–147.

  46. N. Yan, A posteriori error estimators of gradient recovery type for elliptic obstacle problems, Adv. Comput. Math. 15 (2001) 333–362.

    Article  MATH  MathSciNet  Google Scholar 

  47. N. Yan and A. Zhou, Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes, Comput. Methods Appl. Mech. Engrg. 190 (2001) 4289–4399.

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Zhou, C.L. Liem, T.M. Shih and T. Lü, Error analysis on bi-parameter finite elements, Comput. Methods Appl. Mech. Engrg. 158 (1998) 329–339.

    Article  MathSciNet  MATH  Google Scholar 

  49. O.C. Zienkiewicz and J.Z. Zhu, The supercovergent patch recovery and a posteriori error estimates, Internat. J. Numer. Methods Engrg. 33 (1992) 1331–1382.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, D-09107, Chemnitz, Germany

    Reinhold Schneider

  2. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA, Beijing, 100080, P. R. China

    Yuesheng Xu

  3. Institute of Computational Mathematics, and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080, P. R. China

    Aihui Zhou

Authors
  1. Reinhold Schneider
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  2. Yuesheng Xu
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  3. Aihui Zhou
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Corresponding author

Correspondence to Reinhold Schneider.

Additional information

Communicated by B.-Y. Guo

Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem

Mathematics subject classifications (2000)

65N12, 65N15, 65N30.

Aihui Zhou: Subsidized by the Special Funds for Major State Basic Research Projects, and also partially supported by National Science Foundation of China.

Reinhold Schneider: Supported in part by DFG Sonderforschungsbereich SFB 393.

Yuesheng Xu: Correspondence author. Supported in part by the US National Science Foundation under grants DMS-9973427 and CCR-0312113, by Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under program “Hundreds Distinguished Young Chinese Scientists”.

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Schneider, R., Xu, Y. & Zhou, A. An analysis of discontinuous Galerkin methods for elliptic problems. Adv Comput Math 25, 259–286 (2006). https://doi.org/10.1007/s10444-004-7619-y

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  • DOI: https://doi.org/10.1007/s10444-004-7619-y

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