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Error estimators for nonconforming finite element approximations of the Stokes problem. (English) Zbl 0827.76042

Summary: We define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements, we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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[1] Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276 – 1290. · Zbl 0696.73040 · doi:10.1137/0726074
[2] Ivo Babuška, Ricardo Durán, and Rodolfo Rodríguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements, SIAM J. Numer. Anal. 29 (1992), no. 4, 947 – 964. · Zbl 0759.65069 · doi:10.1137/0729058
[3] I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1 – 40. · Zbl 0593.65064 · doi:10.1016/0045-7825(87)90114-9
[4] I. Babuška and W. C. Rheinboldt, A posteriori error estimators in the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978), 1587-1615. · Zbl 0396.65068
[5] R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283 – 301. · Zbl 0569.65079
[6] Randolph E. Bank and Bruno D. Welfert, A posteriori error estimates for the Stokes equations: a comparison, Comput. Methods Appl. Mech. Engrg. 82 (1990), no. 1-3, 323 – 340. Reliability in computational mechanics (Austin, TX, 1989). · Zbl 0725.65106 · doi:10.1016/0045-7825(90)90170-Q
[7] Randolph E. Bank and Bruno D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. 28 (1991), no. 3, 591 – 623. · Zbl 0731.76040 · doi:10.1137/0728033
[8] 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. · Zbl 0302.65087
[9] E. A. Dari, R. Durán, C. Padra, and V. Vampa, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. (to appear). · Zbl 0853.65110
[10] M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles, Internat. J. Numer. Methods Engrg. 19 (1983), no. 4, 505 – 520. · Zbl 0514.73068 · doi:10.1002/nme.1620190405
[11] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077
[12] A. K. Noor and I. Babuška, Quality assessment and control of finite element solutions, Finite Elem. in Anal. & Design 3 (1987), 1-26. · Zbl 0608.73072
[13] María-Cecilia Rivara, Mesh refinement processes based on the generalized bisection of simplices, SIAM J. Numer. Anal. 21 (1984), no. 3, 604 – 613. · Zbl 0574.65133 · doi:10.1137/0721042
[14] L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483 – 493. · Zbl 0696.65007
[15] R. Stenberg and D. Baroudi, A new nonconforming finite element method for incompressible elasticity, Proc. IVth Finnish Mechanics Days, Lappeenranta, June 5-6, 1991 (to appear).
[16] R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), 309-325. · Zbl 0674.65092
[17] R. Verfürth, A posteriori error estimators for the Stokes equations. II. Nonconforming discretizations, Numer. Math. 60 (1991), no. 2, 235 – 249. · Zbl 0739.76035 · doi:10.1007/BF01385723
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