Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods

Linda El Alaoui; Alexandre Ern

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 6, page 903-929
  • ISSN: 0764-583X

Abstract

top
We analyze residual and hierarchical a posteriori error estimates for nonconforming finite element approximations of elliptic problems with variable coefficients. We consider a finite volume box scheme equivalent to a nonconforming mixed finite element method in a Petrov–Galerkin setting. We prove that all the estimators yield global upper and local lower bounds for the discretization error. Finally, we present results illustrating the efficiency of the estimators, for instance, in the simulation of Darcy flows through heterogeneous porous media.

How to cite

top

Alaoui, Linda El, and Ern, Alexandre. "Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.6 (2004): 903-929. <http://eudml.org/doc/245038>.

@article{Alaoui2004,
abstract = {We analyze residual and hierarchical a posteriori error estimates for nonconforming finite element approximations of elliptic problems with variable coefficients. We consider a finite volume box scheme equivalent to a nonconforming mixed finite element method in a Petrov–Galerkin setting. We prove that all the estimators yield global upper and local lower bounds for the discretization error. Finally, we present results illustrating the efficiency of the estimators, for instance, in the simulation of Darcy flows through heterogeneous porous media.},
author = {Alaoui, Linda El, Ern, Alexandre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite elements; nonconforming methods; a posteriori error estimates; finite volumes; Darcy equations; heterogeneous media; nonconforming methods, a posteriori error estimates; elliptic problems; numerical results; Darcy flows; porous media},
language = {eng},
number = {6},
pages = {903-929},
publisher = {EDP-Sciences},
title = {Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods},
url = {http://eudml.org/doc/245038},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Alaoui, Linda El
AU - Ern, Alexandre
TI - Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 6
SP - 903
EP - 929
AB - We analyze residual and hierarchical a posteriori error estimates for nonconforming finite element approximations of elliptic problems with variable coefficients. We consider a finite volume box scheme equivalent to a nonconforming mixed finite element method in a Petrov–Galerkin setting. We prove that all the estimators yield global upper and local lower bounds for the discretization error. Finally, we present results illustrating the efficiency of the estimators, for instance, in the simulation of Darcy flows through heterogeneous porous media.
LA - eng
KW - finite elements; nonconforming methods; a posteriori error estimates; finite volumes; Darcy equations; heterogeneous media; nonconforming methods, a posteriori error estimates; elliptic problems; numerical results; Darcy flows; porous media
UR - http://eudml.org/doc/245038
ER -

References

top
  1. [1] B. Achchab, S. Achchab and A. Agouzal, Hierarchical robust a posteriori error estimator for a singularly pertubed problem. C.R Acad. Paris I 336 (2003) 95–100. Zbl1028.65116
  2. [2] B. Achchab, A. Agouzal, J. Baranger and J.F. Maitre, Estimateur d’erreur a posteriori hiérarchique. Application aux éléments finis mixtes. Numer. Math. 80 (1998) 159–179. Zbl0909.65076
  3. [3] Y. Achdou and C. Bernardi, Un schéma de volumes ou éléments finis adaptatif pour les équations de Darcy à perméabilité variable. C.R Acad. Paris I 333 (2001) 693–698. Zbl0996.65123
  4. [4] Y. Achdou, C. Bernardi and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96 (2003) 17–42. Zbl1050.76035
  5. [5] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Wiley-Interscience Publication (2000). Zbl1008.65076MR1885308
  6. [6] L. Angermann, A posteriori error estimates for FEM with violated Galerkin orthogonality. Numer. Methods Partial Differential Equations 18 (2002) 241–259. Zbl1003.65058
  7. [7] D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7–32. Zbl0567.65078
  8. [8] R. Bank and K. Smith, A posteriori estimates based on hierarchical bases. SIAM J. Numer. Anal. 30 (1991) 921–935. Zbl0787.65078
  9. [9] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985) 283–301. Zbl0569.65079
  10. [10] R. Becker, P. Hansbo and M.G. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Engrg. 192 (2003) 723–733. Zbl1042.65083
  11. [11] C. Bernardi, private communication. 
  12. [12] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579–608. Zbl0962.65096
  13. [13] D. Braess, Finite elements. Cambridge Univ. Press (1997). Zbl0894.65054MR1463151
  14. [14] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comp. 66 (1997) 465–476. Zbl0864.65068
  15. [15] C. Carstensen and A. Funken, A posteriori error control in low-order finite element discretizations of incompressible stationary flow problems. Math. Comp. 70 (2000) 1353–1381. Zbl1014.76042
  16. [16] B. Courbet and J.-P. Croisille, Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér. 32 (1998) 631–649. Zbl0920.65065
  17. [17] J.-P. Croisille, Finite volume box schemes and mixed methods. ESAIM: M2AN 31 (2000) 1087–1106. Zbl0966.65082
  18. [18] J.-P. Croisille and I. Greff, Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations 8 (2002) 355–373. Zbl1004.65113
  19. [19] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming mixed finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 3 (1973) 33–75. Zbl0302.65087
  20. [20] E. Dari, R. Durán and C. Parda, Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64 (1995) 1017–1033. Zbl0827.76042
  21. [21] E. Dari, R. Durán, C. Parda and V. Vampa, A posteriori error estimators for nonconforming finite element methods. RAIRO Modél Math. Anal. Numér. 30 (1996) 385–400. Zbl0853.65110
  22. [22] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Appl. Math. Ser., Springer, New York 159 (2004). Zbl1059.65103MR2050138
  23. [23] M. Fortin and M. Soulié, A non-conforming piecewise quadratic finite element on triangles. Int. J. Num. Meth. Engrg. 19 (1983) 505–520. Zbl0514.73068
  24. [24] R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for non-conforming finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 237–263. Zbl0843.65075
  25. [25] V. John, A posteriori L 2 -error estimates for the nonconforming P 1 / P 0 -finite element discretization of the Stokes equations. J. Comput. Appl. Math. 96 (1998) 99–116. Zbl0930.65123
  26. [26] G. Kanschat and F.-T. Suttmeier, A posteriori error estimates for non-conforming finite element schemes. Calcolo 36 (1999) 129–141. Zbl0936.65128
  27. [27] O. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. Zbl1058.65120
  28. [28] P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, E. Magenes and I. Galligani Eds., Springer-Verlag, New York, Lect. Notes Math. 606 (1977). Zbl0362.65089MR483555
  29. [29] F. Schieweck, A posteriori error estimates with post-processing for nonconforming finite elements. ESAIM: M2AN 36 (2002) 489–503. Zbl1041.65083
  30. [30] J.-M. Thomas and D. Trujillo, Mixed finite volume methods. Int. J. Num. Meth. Engrg. 46 (1999) 1351–1366. Zbl0948.65125
  31. [31] R. Verfürth, A posteriori error estimators for the Stokes equations. II. Non-conforming discretizations. Numer. Math. 60 (1991) 235–249. Zbl0739.76035
  32. [32] R. Verfürth, A review of a posteriori error estimation and adaptative mesh-refinement techniques. Chichester, England (1996). Zbl0853.65108
  33. [33] B.I. Wohlmuth and R.H.W. Hoppe, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas. Math. Comp. 68 (1999) 1347–1378. Zbl0929.65094

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.