Weighted inf-sup condition and pointwise error estimates for the Stokes problem
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- by Ricardo G. Durán and Ricardo H. Nochetto PDF
- Math. Comp. 54 (1990), 63-79 Request permission
Abstract:
Convergence of mixed finite element approximations to the Stokes problem in the primitive variables is examined in maximum norm. Quasioptimal pointwise error estimates are derived for discrete spaces satisfying a weighted inf-sup condition similar to the Babuška -Brezzi condition. The usual techniques employed to prove the inf-sup condition in energy norm can be easily extended to the present situation, thus providing several examples to our abstract framework. The popular Taylor-Hood finite element is the most relevant one.References
- D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337–344 (1985). MR 799997, DOI 10.1007/BF02576171
- Douglas N. Arnold, L. Ridgway Scott, and Michael Vogelius, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 2, 169–192 (1989). MR 1007396
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
- M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math. 33 (1979), no. 2, 211–224. MR 549450, DOI 10.1007/BF01399555
- Christine Bernardi and Geneviève Raugel, Analysis of some finite elements for the Stokes problem, Math. Comp. 44 (1985), no. 169, 71–79. MR 771031, DOI 10.1090/S0025-5718-1985-0771031-7
- J. M. Boland and R. A. Nicolaides, Stability of finite elements under divergence constraints, SIAM J. Numer. Anal. 20 (1983), no. 4, 722–731. MR 708453, DOI 10.1137/0720048
- F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- 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 R. G. Durán and R. H. Nochetto, Pointwise accuracy of a Petrov-Galerkin approximation to the Stokes problem, SIAM J. Numer. Anal. 26 (1989) (to appear).
- R. Durán, R. H. Nochetto, and Jun Ping Wang, Sharp maximum norm error estimates for finite element approximations of the Stokes problem in $2$-D, Math. Comp. 51 (1988), no. 184, 491–506. MR 935076, DOI 10.1090/S0025-5718-1988-0935076-9
- R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753, DOI 10.1051/m2an/1980140302491
- Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341–354, iii (English, with French summary). MR 464543, DOI 10.1051/m2an/1977110403411
- Lucia Gastaldi and Ricardo H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103–128 (English, with French summary). MR 1015921, DOI 10.1051/m2an/1989230101031
- 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
- R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Analysis 21 (1976), no. 4, 397–431. MR 0404849, DOI 10.1016/0022-1236(76)90035-5
- Frank Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), no. 1, 67–77 (German, with English summary). MR 474884, DOI 10.1007/BF01419529
- Joachim Nitsche, $L_{\infty }$-convergence of finite element approximations, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 261–274. MR 0488848
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
- C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Internat. J. Comput. & Fluids 1 (1973), no. 1, 73–100. MR 339677, DOI 10.1016/0045-7930(73)90027-3 R. Temam, Navier-Stokes equations, North Holland, Amsterdam, 1984.
- R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO Anal. Numér. 18 (1984), no. 2, 175–182. MR 743884, DOI 10.1051/m2an/1984180201751
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 54 (1990), 63-79
- MSC: Primary 65N30; Secondary 65N15, 76-08, 76D07
- DOI: https://doi.org/10.1090/S0025-5718-1990-0995211-2
- MathSciNet review: 995211