Applications of Mathematics

, Volume 57, Issue 6, pp 551–568 | Cite as

New results concerning the DWR method for some nonconforming FEM

Article
First Online:
Received:

Abstract

This paper presents a unified framework for the dual-weighted residual (DWR) method for a class of nonconforming FEM. Our approach is based on a modification of the dual problem and uses various ideas from literature which are combined in a new manner. The results are new error identities for some nonconforming FEM. Additionally, a posteriori error estimates with respect to the discrete H 1-seminorm are derived.

Keywords

nonconforming finite elements dual-weighted residual method a posteriori error estimate 

MSC 2010

65N15 65N30 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Ainsworth: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42 (2005), 2320–2341.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    M. Ainsworth: A posteriori error estimation for non-conforming quadrilateral finite elements. Int. J. Numer. Anal. Model. 2 (2005), 1–18.MathSciNetMATHGoogle Scholar
  3. [3]
    M. Ainsworth, R. Rankin: Fully computable bounds for the error in nonconforming finite element approximations of arbitrary order on triangular elements. SIAM J. Numer. Anal. 46 (2008), 3207–3232.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    M. Ainsworth, T. Vejchodský: Fully computable robust a posteriori error bounds for singularly perturbed reaction-diffusion problems. Numer. Math. 119 (2011), 219–243.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    R. Becker, R. Rannacher: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10 (2001), 1–102.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    C. Carstensen, J. Hu, A. Orlando: Framework for the a posteriori error analysis of nonconforming finite elements. SIAM J. Numer. Anal. 45 (2007), 68–82.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    C. Carstensen, J. Hu: A unifying theory of a posteriori error control of nonconforming finite element methods. Numer. Math. 107 (2007), 473–502.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    E. Dari, R. Duran, C. Padra, V. Vampa: A posteriori error estimators for nonconforming finite element methods. RAIRO, Modélisation Math. Anal. Numér. 30 (1996), 385–400.MathSciNetMATHGoogle Scholar
  9. [9]
    J. Douglas Jr., J. E. Santos, D. Sheen, X. Ye: Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. M2AN, Math. Model. Numer. Anal. 33 (1999), 747–770.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    M. Grajewski, J. Hron, S. Turek: Dual weighted a posteriori error estimation for a new nonconforming linear finite element on quadrilaterals. Appl. Numer. Math. 54 (2005), 504–518.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    H.-D. Han: A finite element approximation of Navier-Stokes equations using nonconforming elements. J. Comput. Math. 2 (1984), 77–88.MATHGoogle Scholar
  12. [12]
    G. Kanschat, F.-T. Suttmeier: A posteriori error estimates for nonconforming finite element schemes. Calcolo 36 (1999), 129–141.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    R. Rannacher, S. Turek: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97–111.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    F. Schieweck: A posteriori error estimates with post-processing for nonconforming finite elements. M2AN, Math. Model. Numer. Anal. 36 (2002), 489–503.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.Institut für Numerische MathematikTechnische UniversitätDresdenGermany

Personalised recommendations