Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow
Authors:
W. Dörfler and M. Ainsworth
Journal:
Math. Comp. 74 (2005), 1599-1619
MSC (2000):
Primary 65N12, 65N15, 65N30
DOI:
https://doi.org/10.1090/S0025-5718-05-01743-6
Published electronically:
January 3, 2005
MathSciNet review:
2164088
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We derive computable a posteriori error estimates for the lowest order nonconforming Crouzeix–Raviart element applied to the approximation of incompressible Stokes flow. The estimator provides an explicit upper bound that is free of any unknown constants, provided that a reasonable lower bound for the inf-sup constant of the underlying problem is available. In addition, it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant.
- A. Agouzal, A posteriori error estimator for nonconforming finite element methods, Appl. Math. Lett. 7 (1994), no. 5, 61–66. MR 1350612, DOI https://doi.org/10.1016/0893-9659%2894%2990074-4
- M. Ainsworth, Robust a posteriori error estimates for non-conforming finite element approximation, SIAM J. Numer. Anal., to appear.
- Thomas Apel, Serge Nicaise, and Joachim Schöberl, A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges, IMA J. Numer. Anal. 21 (2001), no. 4, 843–856. MR 1867421, DOI https://doi.org/10.1093/imanum/21.4.843
- Weizhu Bao and John W. Barrett, A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-Newtonian flow, RAIRO Modél. Math. Anal. Numér. 32 (1998), no. 7, 843–858 (English, with English and French summaries). MR 1654432, DOI https://doi.org/10.1051/m2an/1998320708431
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
- Carsten Carstensen and Stefan A. Funken, Fully reliable localized error control in the FEM, SIAM J. Sci. Comput. 21 (1999/00), no. 4, 1465–1484. MR 1742328, DOI https://doi.org/10.1137/S1064827597327486
- Carsten Carstensen and Stefan A. Funken, A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems, Math. Comp. 70 (2001), no. 236, 1353–1381. MR 1836908, DOI https://doi.org/10.1090/S0025-5718-00-01264-3
- Evgenii V. Chizhonkov and Maxim A. Olshanskii, On the domain geometry dependence of the LBB condition, M2AN Math. Model. Numer. Anal. 34 (2000), no. 5, 935–951. MR 1837762, DOI https://doi.org/10.1051/m2an%3A2000110
- M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
- Enzo Dari, Ricardo Durán, and Claudio Padra, Error estimators for nonconforming finite element approximations of the Stokes problem, Math. Comp. 64 (1995), no. 211, 1017–1033. MR 1284666, DOI https://doi.org/10.1090/S0025-5718-1995-1284666-9
- E. Dari, R. G. Durán, and C. Padra, Maximum norm error estimators for three-dimensional elliptic problems, SIAM J. Numer. Anal. 37 (2000), no. 2, 683–700. MR 1740762, DOI https://doi.org/10.1137/S0036142998340253
- E. Dari, R. Duran, C. Padra, and V. Vampa, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 4, 385–400 (English, with English and French summaries). MR 1399496, DOI https://doi.org/10.1051/m2an/1996300403851
- Qing-ping Deng, Xue-jun Xu, and Shu-min Shen, Maximum norm error estimates of Crouzeix-Raviart nonconforming finite element approximation of Navier-Stokes problem, J. Comput. Math. 18 (2000), no. 2, 141–156. MR 1750943
- Ricardo Durán and Claudio Padra, An error estimator for nonconforming approximations of a nonlinear problem, Finite element methods (Jyväskylä, 1993) Lecture Notes in Pure and Appl. Math., vol. 164, Dekker, New York, 1994, pp. 201–205. MR 1299990, DOI https://doi.org/10.1201/b16924-18
- M. Fortin, Utilisation de la méthode des éléments finis en mécanique des fluides. I, Calcolo 12 (1975), no. 4, 405–441 (French, with English summary). MR 421339, DOI https://doi.org/10.1007/BF02575757
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205
- D. F. Griffiths, Finite elements for incompressible flow, Math. Methods Appl. Sci. 1 (1979), no. 1, 16–31. MR 548403, DOI https://doi.org/10.1002/mma.1670010103
- F. Hecht, Construction d’une base de fonctions $P_{1}$ non conforme à divergence nulle dans ${\bf R}^{3}$, RAIRO Anal. Numér. 15 (1981), no. 2, 119–150 (French, with English summary). MR 618819
- Claudio Padra, A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows, SIAM J. Numer. Anal. 34 (1997), no. 4, 1600–1615. MR 1461798, DOI https://doi.org/10.1137/S0036142994278322
- Marius Paraschivoiu and Anthony T. Patera, A posteriori bounds for linear functional outputs of Crouzeix-Raviart finite element discretizations of the incompressible Stokes problem, Internat. J. Numer. Methods Fluids 32 (2000), no. 7, 823–849. MR 1752470, DOI https://doi.org/10.1002/%28SICI%291097-0363%2820000415%2932%3A7%3C823%3A%3AAID-FLD991%3E3.3.CO%3B2-1
- Roger Pierre, Local mass conservation and $C^0$-discretizations of the Stokes problem, Houston J. Math. 20 (1994), no. 1, 115–127. MR 1272565
- Vitoriano Ruas Santos, Finite element solution of $3$D viscous flow problems using nonstandard degrees of freedom, Japan J. Appl. Math. 2 (1985), no. 2, 415–431. MR 839337, DOI https://doi.org/10.1007/BF03167084
- V. Ruas, Circumventing discrete Korn’s inequalities in convergence analyses of nonconforming finite element approximations of vector fields, Z. Angew. Math. Mech. 76 (1996), no. 8, 483–484. MR 1409361, DOI https://doi.org/10.1002/zamm.19960760813
- F. Schieweck and L. Tobiska, An optimal order error estimate for an upwind discretization of the Navier-Stokes equations, Numer. Methods Partial Differential Equations 12 (1996), no. 4, 407–421. MR 1396464, DOI https://doi.org/10.1002/%28SICI%291098-2426%28199607%2912%3A4%3C407%3A%3AAID-NUM1%3E3.0.CO%3B2-Q
- G. Stoyan, Towards discrete Velte decompositions and narrow bounds for inf-sup constants, Comput. Math. Appl. 38 (1999), no. 7-8, 243–261. MR 1713178, DOI https://doi.org/10.1016/S0898-1221%2899%2900254-0
- R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309–325. MR 993474, DOI https://doi.org/10.1007/BF01390056
- ---, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, 1996.
Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 65N15, 65N30
Retrieve articles in all journals with MSC (2000): 65N12, 65N15, 65N30
Additional Information
W. Dörfler
Affiliation:
Institut für Angewandte Mathematik II, Univ. Karlsruhe, 76128 Karlsruhe, Germany
Email:
doerfler@mathematik.uni-karlsruhe.de
M. Ainsworth
Affiliation:
Department of Mathematics, Strathclyde University, 26 Richmond St., Glasgow G1 1XH, Scotland
MR Author ID:
261514
Email:
M.Ainsworth@strath.ac.uk
Keywords:
Computable error bounds,
a posteriori error estimates,
nonconforming finite elements,
Stokes flow.
Received by editor(s):
November 17, 2003
Received by editor(s) in revised form:
August 7, 2004
Published electronically:
January 3, 2005
Additional Notes:
This work was initiated during the authors’ visit to the Newton Institute for Mathematical Sciences in Cambridge. The support of the second author by the Leverhulme Trust under a Leverhulme Trust Fellowship is gratefully acknowledged.
Article copyright:
© Copyright 2005
American Mathematical Society
