Error estimators for nonconforming finite element approximations of the Stokes problem
HTML articles powered by AMS MathViewer
- by Enzo Dari, Ricardo Durán and Claudio Padra PDF
- Math. Comp. 64 (1995), 1017-1033 Request permission
Abstract:
In this paper 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.References
- 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. MR 1025088, DOI 10.1137/0726074
- 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. MR 1173179, DOI 10.1137/0729058
- 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. MR 880421, DOI 10.1016/0045-7825(87)90114-9 I. Babuška and W. C. Rheinboldt, A posteriori error estimators in the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978), 1587-1615.
- R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283–301. MR 777265, DOI 10.1090/S0025-5718-1985-0777265-X
- 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). MR 1077660, DOI 10.1016/0045-7825(90)90170-Q
- 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. MR 1098409, DOI 10.1137/0728033
- 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 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).
- M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles, Internat. J. Numer. Methods Engrg. 19 (1983), no. 4, 505–520. MR 702056, DOI 10.1002/nme.1620190405
- 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. MR 851383, DOI 10.1007/978-3-642-61623-5 A. K. Noor and I. Babuška, Quality assessment and control of finite element solutions, Finite Elem. in Anal. & Design 3 (1987), 1-26.
- María-Cecilia Rivara, Mesh refinement processes based on the generalized bisection of simplices, SIAM J. Numer. Anal. 21 (1984), no. 3, 604–613. MR 744176, DOI 10.1137/0721042
- L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7 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). R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), 309-325.
- R. Verfürth, A posteriori error estimators for the Stokes equations. II. Nonconforming discretizations, Numer. Math. 60 (1991), no. 2, 235–249. MR 1133581, DOI 10.1007/BF01385723
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp. 64 (1995), 1017-1033
- MSC: Primary 65N30; Secondary 65N50, 76D07, 76M10
- DOI: https://doi.org/10.1090/S0025-5718-1995-1284666-9
- MathSciNet review: 1284666