Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A comparison of a posteriori error estimators
for mixed finite element discretizations
by Raviart-Thomas elements

Authors: Barbara I. Wohlmuth and Ronald H. W. Hoppe
Journal: Math. Comp. 68 (1999), 1347-1378
MSC (1991): Primary 65F10, 65N30, 65N50, 65N55
Published electronically: May 19, 1999
MathSciNet review: 1651760
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

References [Enhancements On Off] (What's this?)

  • 1. B. ACHCHAB, A.AGOUZAL, J. BARANGER, AND J. MAITRE, Estimateur d'erreur a posteriori hiérarchique. Application aux éléments finis mixtes, Numer. Math. 80 (1998), 159-179. CMP 49:01
  • 2. D. ARNOLD AND F. BREZZI, Mixed and nonconforming finite element methods: Implementation, post-processing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), pp. 1347-1378. MR 87g:65176
  • 3. I. BABUKA AND W. RHEINBOLDT, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 1347-1378. MR 58:3400
  • 4. -, A posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng., 12 (1978), pp. 1347-1378.
  • 5. R. BANK, A. SHERMAN, AND A. WEISER, Refinement algorithm and data structures for regular local mesh refinement, in Scientific Computing, R. Stepleman et al., ed., vol. 44, IMACS North-Holland, Amsterdam, 1983, pp. 3-17. MR 85i:00014
  • 6. R. BANK AND A. WEISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283-301. MR 86g:65207
  • 7. R. BECKER AND R. RANNACHER, Weighted a posteriori error control in FE methods, Tech. Report 96 - 1, SFB 359 - Universität Heidelberg, January 1996.
  • 8. F. BORNEMANN, B. ERDMANN, AND R. KORNHUBER, A posteriori error estimates for elliptic problems in two and three spaces dimensions, SIAM J. Numer. Anal., 33 (1996), pp. 1347-1378 MR 98a:65161
  • 9. D. BRAESS, O. KLAAS, R. NIEKAMP, E. STEIN, AND F. WOBSCHAL, Error indicators for mixed finite elements in 2-dimensional linear elasticity, Comput. Methods Appl. Mech. Eng., 127, (1995), pp. 1347-1378 MR 96h:73045
  • 10. D. BRAESS AND R. VERFÜRTH, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal., 33 (1996), pp. 2431-2444 MR 97m:65201
  • 11. J. BRANDTS, Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math., 68 (1994), pp. 1347-1378. MR 97a:65162
  • 12. F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991. MR 92d:65187
  • 13. P. CLÉMENT, Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 2 (1975), pp. 1347-1378. MR 53:4569
  • 14. P. DEUFLHARD, P. LEINEN, AND H. YSERENTANT, Concepts of an adaptive hierarchical finite element code, IMPACT Comput. Sci. Engrg., 1 (1989), pp. 3-35.
  • 15. R. DURAN AND R. RODRIGUEZ, On the asymptotic exactness of Bank-Weiser's estimator, Numer. Math., 62 (1992), pp. 1347-1378. MR 93e:65135
  • 16. K. ERIKSSON AND C. JOHNSON, An adaptive finite element method for linear elliptic problems, Math. Comput., 50 (1988), pp. 1347-1378.
  • 17. R. EWING AND J. WANG, Analysis of multilevel decomposition iterative methods for mixed finite element methods, RAIRO Modél. Math. Anal. Numér. 28 (1994), pp. 1347-1378. MR 95e:65099
  • 18. R. HOPPE AND B. WOHLMUTH, Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems, SIAM J. Numer. Anal., 34 (1997), pp. 1347-1378. MR 98e:65095
  • 19. -, Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods, Appl. Math., 40 (1995), pp. 1347-1378. MR 96b:65111
  • 20. -, Multilevel iterative solution and adaptive mesh refinement for mixed finite element discretizations, Appl. Num. Math., 23 (1997), pp. 1347-1378 MR 98g:65109
  • 21. -, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), pp. 1347-1378. MR 97e:65124
  • 22. C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, no. 101, Cambridge University Press, Cambridge, 1987. MR 89b:65003a
  • 23. C. JOHNSON AND P. HANSBO, Adaptive finite element methods in computational mechanics, Comp. Meth. Appl. Mech. Eng., 101 (1992), pp. 143-181. MR 93m:65157
  • 24. R. RODRIGUEZ, Some remarks on the Zienkiewicz-Zhu estimator, Numer. Meth. for PDE, 10 (1994), pp. 1347-1378. MR 95e:65103
  • 25. T. STROUBOULIS AND J. ODEN, A posteriori error estimation of finite element approximations in fluid dynamics, Comp. Meth. Appl. Mech. Engrg., 78 (1990), pp. 1347-1378. MR 91a:76066
  • 26. B. SZABÓ AND I. BABUKA, Finite Element Analysis, Wiley, New York, 1991. MR 93f:73001
  • 27. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, 1996.
  • 28. -, A posteriori error estimators for the Stokes equations, Numer. Math., 55 (1989), pp. 1347-1378. MR 90d:65187
  • 29. -, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comput., 62 (1994), pp. 1347-1378. MR 94j:65136
  • 30. -, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comp. Appl. Math., 50 (1994), pp. 1347-1378. MR 95c:65171
  • 31. B. WOHLMUTH, Adaptive Multilevel-Finite-Elemente Methoden zur Lösung elliptischer Randwertprobleme, PhD thesis, TU München, 1995.
  • 32. J. ZHU AND O. ZIENKIEWICZ, Adaptive techniques in the finite element method, Commun. Appl. Numer. Methods, 4 (1988), pp. 1347-1378.
  • 33. O. ZIENKIEWICZ AND R. TAYLOR, The Finite Element Method, vol. 1, Mc Graw-Hill, London, 1989.
  • 34. O. ZIENKIEWICZ AND J. ZHU, A simple error estimator and adaptive procedure for practical engineering analysis, J. Numer. Meth. Eng., 28 (1987), pp. 1347-1378.

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65F10, 65N30, 65N50, 65N55

Retrieve articles in all journals with MSC (1991): 65F10, 65N30, 65N50, 65N55

Additional Information

Barbara I. Wohlmuth
Affiliation: Math. Institute, University of Augsburg, D-86135 Augsburg, Germany

Ronald H. W. Hoppe
Affiliation: Math. Institute, University of Augsburg, D-86135 Augsburg, Germany

Keywords: Mixed finite elements, a posteriori error estimation, adaptive grid refinement
Published electronically: May 19, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society