Skip to main content
SpringerLink
Log in

Mixed Finite Element Methods

  • Chapter
Mixed Finite Elements, Compatibility Conditions, and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1939))

The mathematical analysis and applications of mixed finite element methods have been widely developed since the seventies. A general analysis for this kind of methods was first developed by Brezzi [13]. We also have to mention the papers by Babuška [9] and by Crouzeix and Raviart [22] which, although for particular problems, introduced some of the fundamental ideas for the analysis of mixed methods. We also refer the reader to [32, 31], where general results were obtained, and to the books [17, 45, 37].

The rest of this work is organized as follows: in Sect. 2 we review some basic tools for the analysis of finite element methods. Section 3 deals with the mixed formulation of second order elliptic problems and their finite element approximation. We introduce the Raviart–Thomas spaces [44, 49, 41] and their generalization to higher dimensions, prove some of their basic properties, and construct the Raviart–Thomas interpolation operator which is a basic tool for the analysis of mixed methods. Then, we prove optimal order error estimates and a superconvergence result for the scalar variable. We follow the ideas developed in several papers (see for example [24, 16]). Although for simplicity we consider the Raviart–Thomas spaces, the error analysis depends only on some basic properties of the spaces and the interpolation operator, and therefore, analogous results hold for approximations obtained with other finite element spaces. We end the section recalling other known families of spaces and giving some references. In Sect. 4 we introduce an a posteriori error estimator and prove its equivalence with an appropriate norm of the error up to higher order terms. For simplicity, we present the a posteriori error analysis only in the 2-d case. Finally, in Sect. 5, we introduce the general abstract setting for mixed formulations and prove general existence and approximation results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Acosta and R. G. Durán, The maximum angle condition for mixed and non conforming elements: Application to the Stokes equations, SIAM J. Numer. Anal. 37, 18–36, 2000.

    Article  Google Scholar 

  2. G. Acosta, R. G. Durán and M. A. Muschietti, Solutions of the divergence operator on John domains, Adv. Math. 206, 373–401, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Ainsworth, Robust a posteriori error estimation for nonconforming finite element approximations, SIAM J. Numer. Anal. 42, 2320–2341, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Alonso, Error estimator for a mixed method, Numer. Math. 74, 385–395, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. N. Arnold, D. Boffi and R. S. Falk, Quadrilateral H({div}) finite elements, SIAM J. Numer. Anal. 42, 2429–2451, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods implementation, postprocessing and error estimates, R.A.I.R.O., Modél. Math. Anal. Numer. 19, 7–32, 1985.

    MATH  MathSciNet  Google Scholar 

  7. D. N. Arnold, L. R. Scott and M. Vogelius, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon, Ann. Scuola Norm. Sup. Pisa Cl. Sci-Serie IV, XV, 169–192, 1988.

    MathSciNet  Google Scholar 

  8. I. Babuška, Error bounds for finite element method, Numer. Math. 16, 322–333, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  9. I. Babuška, The finite element method with lagrangian multipliers, Numer. Math., 20, 179–192, 1973.

    Article  MATH  Google Scholar 

  10. A. Bermúdez, P. Gamallo, M. R. Nogueiras and R. Rodríguez, Approximation of a structural acoustic vibration problem by hexhaedral finite elements, IMA J. Numer. Anal. 26, 391–421, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. H. Bramble and J. M. Xu, Local post-processing technique for improving the accuracy in mixed finite element approximations, SIAM J. Numer. Anal. 26, 1267–1275, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Brenner and L. R. Scott, The Mathematical Analysis of Finite Element Methods, Springer, Berlin Heidelberg New York, 1994.

    Google Scholar 

  13. F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from lagrangian multipliers, R.A.I.R.O. Anal. Numer. 8, 129–151, 1974.

    MathSciNet  Google Scholar 

  14. F. Brezzi, J. Douglas, R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51, 237–250, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Brezzi, J. Douglas, M. Fortin and L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables, Math. Model. Numer. Anal. 21, 581–604, 1987.

    MathSciNet  Google Scholar 

  16. F. Brezzi, J. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47, 217–235, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Brezzi, and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin Heidelberg New York, 1991.

    MATH  Google Scholar 

  18. C. Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66, 465–476, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  19. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1978.

    Google Scholar 

  20. P. G. Ciarlet, Mathematical Elasticity, Volume 1. Three-Dimensional Elasticity, North Holland, 1988.

    Google Scholar 

  21. P. Clément, Approximation by finite element function using local regularization, RAIRO R-2 77–84, 1975.

    Google Scholar 

  22. M. Crouzeix and P. A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, R.A.I.R.O. Anal. Numer. 7, 33–76, 1973.

    MathSciNet  Google Scholar 

  23. E. Dari, R. G. Durán, C. Padra and V. Vampa, A posteriori error estimators for nonconforming finite element methods, Math. Model. Numer. Anal. 30, 385–400, 1996.

    MATH  Google Scholar 

  24. J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44, 39–52, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  25. T. Dupont, and L. R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34, 441–463, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. G. Durán, On polynomial Approximation in Sobolev Spaces, SIAM J. Numer. Anal. 20, 985–988, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. G. Durán, Error Analysis in L p for Mixed Finite Element Methods for Linear and quasilinear elliptic problems, R.A.I.R.O. Anal. Numér 22, 371–387, 1988.

    MATH  Google Scholar 

  28. R. G. Durán, Error estimates for anisotropic finite elements and applications Proceedings of the International Congress of Mathematicians, 1181–1200, 2006.

    Google Scholar 

  29. R. G. Durán and A. L. Lombardi, Error estimates for the Raviart-Thomas interpolation under the maximum angle condition, preprint, http://mate.dm.uba.ar/~rduran/papers/dl3.pdf

  30. R. G. Durán and M. A. Muschietti, An explicit right inverse of the divergence operator which is continuous in weighted norms, Studia Math. 148, 207–219, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  31. R. S. Falk and J. Osborn, Error estimates for mixed methods, R.A.I.R.O. Anal. Numer. 4, 249–277, 1980.

    MathSciNet  Google Scholar 

  32. M. Fortin, An analysis of the convergence of mixed finite element methods, R.A.I.R.O. Anal. Numer. 11, 341–354, 1977.

    MATH  MathSciNet  Google Scholar 

  33. E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova 27, 284–305, 1957.

    MATH  MathSciNet  Google Scholar 

  34. L. Gastaldi and R. H. Nochetto, Optimal L ∞-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50, 587–611, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  35. L. Gastaldi and R. H. Nochetto, On L ∞- accuracy of mixed finite element methods for second order elliptic problems, Mat. Aplic. Comp. 7, 13–39, 1988.

    MATH  MathSciNet  Google Scholar 

  36. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin Heidelberg New York, 1983.

    MATH  Google Scholar 

  37. V. Girault and P. A. Raviart, Element Methods for Navier–Stokes Equations, Springer, Berlin Heidelberg New York, 1986.

    MATH  Google Scholar 

  38. P. Grisvard, Elliptic Problems in Nonsmooth Domain, Pitman, Boston, 1985.

    Google Scholar 

  39. C. Lovadina and R. Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp. 75, 1659–1674, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  40. L. D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart–Thomas mixed method, SIAM J. Numer. Anal. 22, 493–496, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. C. Nédélec, Mixed finite elements in I​R3, Numer. Math. 35, 315–341, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  42. J. C. Nédélec, A new family of mixed finite elements in I​R3, Numer. Math. 50, 57–81, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  43. L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rat. Mech. Anal. 5, 286–292, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  44. P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, (I. Galligani, E. Magenes, eds.), Lectures Notes in Mathematics, vol. 606, Springer, Berlin Heidelberg New York, 1977.

    Chapter  Google Scholar 

  45. J. E. Roberts and J. M. Thomas, Mixed and Hybrid Methods in Handbook of Numerical Analysis, Vol. II (P. G. Ciarlet and J. L. Lions, eds.), Finite Element Methods (Part 1), North Holland, 1989.

    Google Scholar 

  46. J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements, Preprint ISC-01-10-MATH, Texas A&M University, 2001.

    Google Scholar 

  47. L. R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comp. 54, 483–493, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  48. R. Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO, Model. Math. Anal. Numer. 25, 151–167, 1991.

    MATH  MathSciNet  Google Scholar 

  49. J. M. Thomas, Sur l’Analyse Numérique des Méthodes d’Éléments Finis Hybrides et Mixtes, Thèse de Doctorat d’Etat, Université Pierre et Marie Curie, Paris, 1977.

    Google Scholar 

  50. R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55, 309–325, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  51. R. Verfürth, A note on polynomial approximation in Sobolev spaces, RAIRO M2AN 33, 715–719, 1999.

    Article  MATH  Google Scholar 

  52. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley, New York, 1996.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina

    Ricardo G. Durán

Authors
  1. Ricardo G. Durán
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133, Brescia, Italy

    Lucia Gastaldi

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Durán, R.G. (2008). Mixed Finite Element Methods. In: Boffi, D., Gastaldi, L. (eds) Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol 1939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78319-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation