Skip to main content
Log in

A unifying theory of a posteriori error control for nonconforming finite element methods

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space \(V_h^c\) annulates the linear and bounded residual ℓ written \(V_h^c \subseteq {\rm ker} \ell\) . That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that \(V_h^c \not\subset {\rm ker} \ell\) . The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator \(\Pi: V_h^c \rightarrow V_h^{nc}\) with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lamé equations illustrate the presented unifying theory of a posteriori error control for NCFEM.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ainsworth M. and Oden J.T. (2000). A posteriori error estimation in finite element analysis. Wiley, New York

    MATH  Google Scholar 

  2. Ainsworth M. (2005). A posteriori error estimation for non-conforming quadrilateral finite elements. Int. J. Numer. Anal. Model 2: 1–18

    MATH  MathSciNet  Google Scholar 

  3. Arnold D.N. (1982). An interior penalty finite element method with discontinuous elements. IAM J. Numer. Anal. 19: 742–760

    Article  MATH  Google Scholar 

  4. Arnold D.N., Brezzi F., Cockburn B. and Marini D. (2000). Discontinuous Galerkin methods for elliptic problems. In: Cockburn, B., Karniadakis, G., and Shu, C.W. (eds) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11., pp 89–101. Springer, New York

    Google Scholar 

  5. Baker G. (1977). Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31: 45–59

    Article  MATH  MathSciNet  Google Scholar 

  6. Baker G.A., Jureidini W.N. and Karakashian O.A. (1990). Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Numer. Anal. 27: 1466–1485

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Bernardi C. and Hecht F. (2002). Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comp. 71: 1371–1403

    Article  MATH  MathSciNet  Google Scholar 

  10. Bernardi C., Maday Y. and Patera A.T. (1993). Domain decomposition by the mortar element method. In: Kaper, H. (eds) Asymptotic and Numerical Methods for Partial Differential Equations and Their Applications., pp 269–286. Reidel, Dordrecht

    Google Scholar 

  11. Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Nonlinear Partial Differential Equations and Their Applications, Paris, pp. 13–51 (1994)

  12. Bernardi, C., Owens, R.G. Valenciano, J.: An error indicator for mortar element solutions to the Stokes problem. In: Internal Report 99030, Laboratoire d’Analyse Numrique, Université Pierre et Marie Curie, Paris (1999)

  13. Braess D. (1997). Finite Elements. Cambridge University Press, London

    Google Scholar 

  14. Braess D., Carstensen C. and Reddy B.D. (2004). Uniform convergence and a posteriori error estimators for the enhanced strain finite element method. Numer. Math. 96: 461–479

    Article  MATH  MathSciNet  Google Scholar 

  15. Brenner S.C. and Scott L.R. (2002). The Mathematical Theory of Finite Element Methods. Springer, Berlin

    MATH  Google Scholar 

  16. Brenner S.C. and Sung L.Y. (1992). Linear finite element methods for planar linear elasticity. Math. Comp. 59: 321–338

    Article  MATH  MathSciNet  Google Scholar 

  17. Brezzi F. and Fortin M. (1991). Mixed and Hybrid Finite Element Methods. Springer, Berlin

    MATH  Google Scholar 

  18. Bustinza R. and Gabriel N. (2005). Gatica and Bernardo Cockburn. An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput. 22: 147–185

    Article  MathSciNet  Google Scholar 

  19. Cai Z., Ye X. and Douglas J. Jr. (1999). A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations. CaLcoLo 36: 215–232

    Article  MATH  MathSciNet  Google Scholar 

  20. Carstensen C. (2005). A unifying theory of a posteriori finite element error control. Numer. Math. 100: 617–637

    Article  MATH  MathSciNet  Google Scholar 

  21. Carstensen C. (1999). Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN Math. Model. Numer. Anal. 33: 1187–1202

    Article  MATH  MathSciNet  Google Scholar 

  22. Carstensen C. and Bartels S. (2002). Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming and mixed FEM. Math. Comp. 71: 945–969

    MATH  MathSciNet  Google Scholar 

  23. Carstensen C., Bartels S. and Jansche S. (2002). A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92: 233–256

    Article  MATH  MathSciNet  Google Scholar 

  24. Carstensen C. and Dolzmann G. (1998). A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81: 187–209

    Article  MATH  MathSciNet  Google Scholar 

  25. Carstensen, C., Hu, J., Orlando, A.: Framework for the a posteriori error analysis of nonconforming finite elements. Preprint (2005-11), Department of Mathematics, Humboldt University of Berlin (2005). SIAM J. Numer. Anal. 45, 68–82 (2007)

  26. Creusé E., Kunert G. and Nicaise S. (2004). A posteriori error estimation for the Stokes problem: anisotropic and isotropic discretications. M3AS 14: 1–48

    Google Scholar 

  27. Crouzeix M. and Raviart P.-A. (1973). Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7: 33–76

    MathSciNet  Google Scholar 

  28. Dari E., Duran R. and Padra C. (1995). Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64: 1017–1033

    Article  MATH  MathSciNet  Google Scholar 

  29. Dari E., Duran R., Padra C. and Vampa V. (1996). A posteriori error estimators for nonconforming finite element methods. Math. Model. Numer. Anal. 30: 385–400

    MATH  MathSciNet  Google Scholar 

  30. Douglas J. Jr., Dupont T. (1976) Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Lectures Notes in Physics, vol. 58. Springer, Berlin

  31. Santos J.E., Sheen D., Ye X. and Douglas J. Jr. (1999). Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. Math. Model. Numer. Anal. 33: 747–770

    Article  MATH  MathSciNet  Google Scholar 

  32. Falk R.S. (1991). Nonconforming finite element methods for the equations of linear elasticity. Math. Comp. 57: 529–550

    Article  MATH  MathSciNet  Google Scholar 

  33. Grajewski, M., Hron, J., Turek, S.: Numerical analysis for a new non-conforming linear finite element on quadrilaterals. J. Comput. Appl. Math. (in press)

  34. Grajewski, M., Hron, J., Turek, S.: Dual Weighted a posteriori error estimation for a new nonconforming linear finite element on quadrilaterals. www.mathematik.uni-dortmund.de/lsiii/static/showpdffile_GrajewskiHronTurek2004.pdf

  35. Girault V. and Raviart P.-A. (1986). Finite Element Methods for Navier–Stokes Equations. Springer, Berlin

    MATH  Google Scholar 

  36. Han H.-D. (1984). Nonconforming elements in the mixed finite element method. J. Comp. Math. 2: 223–233

    MATH  Google Scholar 

  37. Houston P., Schotzau D. and Wihler T.P. (2005). Energy norm shape a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. Comput. 22: 347–370

    Article  MathSciNet  Google Scholar 

  38. Houston P., Schotzau D. and Wihler T.P. (2006). An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity. Comp. Methods Appl. Mech. Engrg. 195: 224–3246

    Article  MathSciNet  Google Scholar 

  39. Houston, P., Schotzau, D., Wihler, T.P.: Energy norm a posteriori error estimation of hp- adaptive discontinuous Galerkin methods for elliptic problems. M3AS (to appear)

  40. Hu J., Man H.-Y. and Shi Z.-C. (2005). Constrained nonconforming rotated Q 1 element for Stokes flow and planar elasticity. Math. Numer. Sin. (in Chinese) 27: 311–324

    MathSciNet  Google Scholar 

  41. Hu J. and Shi Z.-C. (2005). Constrained quadrilateral nonconforming rotated Q 1-element. J. Comp. Math. 23: 561–586

    MATH  MathSciNet  Google Scholar 

  42. Kanschat G. and Suttmeier F.-T. (1999). A posteriori error estimates for nonconforming finite element schemes. Calcolo 36: 129–141

    Article  MATH  MathSciNet  Google Scholar 

  43. Karakashian O.A. and Jureidini W.N. (1998). A nonconforming finite element method for the stationary Navier–Stokes equations. SIAM J. Numer. Anal. 35: 93–120

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  45. Kouhia R. and Stenberg R. (1995). A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124: 195–212

    Article  MATH  MathSciNet  Google Scholar 

  46. Lee C.O., Lee J. and Sheen D.W. (2003). A locking-free nonconforming finite element method for planar linear elasticity. Adv. Comput. Math. 19: 277–291

    Article  MATH  MathSciNet  Google Scholar 

  47. Lin Q., Tobiska L. and Zhou A. (2005). On the superconvergence of nonconforming low order finite elements applied to the Poisson equation. IMA. J. Numer. Anal. 25: 160–181

    Article  MATH  MathSciNet  Google Scholar 

  48. Ming, P.-B.: Nonconforming finite element vs locking problem. Doctorate Dissertation (in Chinese), Institute of Computational Mathematics, Chinese Academy of Science (1999)

  49. Park C. and Sheen D. (2003). P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41: 624–640

    Article  MATH  MathSciNet  Google Scholar 

  50. Rannacher R. and Turek S. (1992). Simple nonconforming quadrilateral Stokes element. Numer. Methods PDE 8: 97–111

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  52. Riviere B. and Wheeler M.F. (2003). A posteriori error estimates and mesh adaptation strategy for discontinuous Galerkin methods applied to diffusion problems. Comput. Math. Appl. 46: 141–163

    Article  MATH  MathSciNet  Google Scholar 

  53. Simo J.C. and Rifai M.S. (1990). A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Engrg. 29: 1595–1638

    Article  MATH  MathSciNet  Google Scholar 

  54. Shi Z.-C. (1984). A convergence condition for quadrilateral Wilson element. Numer. Math. 44: 349–361

    Article  MATH  MathSciNet  Google Scholar 

  55. Verfürth R. (1996). A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley–Teubner, Stuttgart

    MATH  Google Scholar 

  56. Wang L.-H. and Qi H. (2002). On locking-free finite element schemes for the pure displacement boundary value problem in the planar elasticity. Math. Numer. Sin. (in Chinese) 24: 243–256

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  58. Wihler T.P. (2006). Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comp. 75: 1087–1102

    Article  MATH  MathSciNet  Google Scholar 

  59. Wilson E.L., Taylor R.L., Doherty W. and Ghaboussi J. (1973). Incompatible displacement models. In: Fenves, S.J., Perrone, N., Robinson, A.R., and Schnobrich, W.C. (eds) Numerical and Computer Methods in Structural Mechanics., pp 43–57. Academic, New York

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  61. Zhang Z.-M. (1997). Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34: 640–663

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. Carstensen.

Additional information

Supported by DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the German Indian Project DST-DAAD (PPP-05). J. Hu was partially supported by National Science Foundation of China under Grant No.10601003.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carstensen, C., Hu, J. A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107, 473–502 (2007). https://doi.org/10.1007/s00211-007-0068-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-007-0068-z

Mathematics Subject Classification (2000)

Navigation