×

zbMATH — the first resource for mathematics

A posteriori error estimators for the Stokes equations. II: Non- conforming discretizations. (English) Zbl 0739.76035
Summary: [For part I, see: the author, ibid. 55, No. 3, 309-325 (1989; Zbl 0674.65092).]
We present an a posteriori error estimator for the nonconforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other nonconforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76D05 Navier-Stokes equations for incompressible viscous fluids
65F10 Iterative numerical methods for linear systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
Software:
FEMFLOW
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Adams, R.A. (1975): Sobolev Spaces. Academic Press, New York · Zbl 0314.46030
[2] Babuska, I., Rheinboldt, W.C. (1978): A posteriori error estimates for the finite element method. Int. J. Numer. Methods in Engrg.12, 1597-1615 · Zbl 0396.65068 · doi:10.1002/nme.1620121010
[3] Babuska, I., Rheinboldt, W.C. (1976): Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, 736-754 · Zbl 0398.65069 · doi:10.1137/0715049
[4] Bank, R.E., Weiser, A. (1985): Some a posteriori error estimators for elliptic partial differential equations. Math. Comput.44, 283-301 · Zbl 0569.65079 · doi:10.1090/S0025-5718-1985-0777265-X
[5] Bank, R.E., Welfert, B.D. (1990): A posteriori error estimates for the Stokes equations: a comparison. Comp. Meth. Appl. Mech. Engrg.82, 323-340 · Zbl 0725.65106 · doi:10.1016/0045-7825(90)90170-Q
[6] Bank, R.E., Welfert, B.D. (1990): A posteriori error estimates for the Stokes problem. Preprint, Univ. of California San Diego · Zbl 0725.65106
[7] Bernardi, C., Raugel, G. (1981): M?thodes d’?l?ments finis mixtes pour les ?quations de Stokes et de Navier-Stokes dans un polygone non-convexe. Calcolo18, 255-291 · Zbl 0475.76035 · doi:10.1007/BF02576359
[8] Braess, D., Verf?rth, R. (1990): Multi-grid methods for non-conforming finite element methods. SIAM J. Numer. Anal.27, 979-986 · Zbl 0703.65067 · doi:10.1137/0727056
[9] Brenner, S. (1988): Multigrid methods for nonconforming finite elements. PhD Thesis, Univ. of Michigan
[10] Brenner, S. (1990): A nonconforming multigrid method for the stationary Stokes equations. Math. Comput.55, 411-437 · Zbl 0705.76027 · doi:10.1090/S0025-5718-1990-1035927-5
[11] Brezzi, F., Rappaz, J., Raviart, P.A. (1980): Finite dimensional approximation of non-linear problems I. Branches of non-singular solutions. Numer. Math.36, 1-25 (1980) · Zbl 0488.65021 · doi:10.1007/BF01395985
[12] Cl?ment, P. (1975): Approximation by finite element functions using local regularization. RAIRO Anal. Num?r.9, 77-84
[13] Crouzeix, M., Raviart, P.A. (1973): Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Num?r.7, 33-76 · Zbl 0302.65087
[14] Dobrowolski, M., Thomas, K. (1984): On the use of discrete solenoidal finite elements for approximating the Navier-Stokes equation. In: Application of Mathematics in Technology, Proc. Ger.-Ital. Symp., Rome pp. 246-262 · Zbl 0547.76039
[15] D?rfler, W. (1990): The condition of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics. Numer. Math.58, 203-214 · Zbl 0705.76026 · doi:10.1007/BF01385619
[16] Eriksson, K., Johnson, C. (1988): An adaptive finite element method for linear elliptic problems. Math. Comput.50, 361-383 · Zbl 0644.65080 · doi:10.1090/S0025-5718-1988-0929542-X
[17] Girault, V., Raviart, P.A. (1986): Finite element approximation of the Navier-Stokes equations. Computational Methods in Physics. Springer, Berlin Heidelberg New York · Zbl 0585.65077
[18] Griffiths, D.F. (1979): Finite elements for incompressible flow. Math. Meth. Appl. Sci.1, 16-31 · Zbl 0425.65061 · doi:10.1002/mma.1670010103
[19] Kellogg, R.B., Osborn, J.E. (1976): A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal.21, 397-431 · Zbl 0317.35037 · doi:10.1016/0022-1236(76)90035-5
[20] Verf?rth, R. (1989): A posteriori error estimators for the Stokes equations. Numer. Math.55, 309-325 · Zbl 0674.65092 · doi:10.1007/BF01390056
[21] Verf?rth, R. (1990): A posteriori error estimators and adaptive mesh-refinement for a mixed finite element discretization of the Navier-Stokes equations. In: W. Hackbusch, R. Rannacher, eds., Proc. 5th GAMM Conf. on Numerical Methods for the Navier-Stokes Equations. Vieweg, Braunschweig, pp. 145-152
[22] Verf?rth, R. (1989): On the preconditioning of non-conforming solenoidal finite element approximations of the Stokes equation. Preprint, Universit?t Z?rich
[23] Verf?rth, R. (1989): FEMFLOW-user guide. Version 1. Preprint, Universit?t Z?rich
[24] Welfert, B.D. (1990): A posteriori error estimates for the Stokes problem. PhD Thesis, Univ. of California San Diego · Zbl 0725.65106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.