Sharp maximum norm error estimates for finite element approximations of the Stokes problem in $2$-D
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- by R. Durán, R. H. Nochetto and Jun Ping Wang PDF
- Math. Comp. 51 (1988), 491-506 Request permission
Abstract:
This paper deals with finite element approximations of the Stokes equations in a plane bounded domain $\Omega$, using the so-called velocity-pressure mixed formulation. Quasi-optimal error estimates in the maximum norm are derived for the velocity, its gradient and the pressure fields. The analysis relies on abstract properties which are in turn a consequence of the eixstence of a local projection operator ${\Pi _h}$ satisfying \[ \int _\Omega \operatorname {div}({\mathbf {v}} - {\Pi _h}{\mathbf {v}})q\;d{\mathbf {x}} = 0,\quad \forall {\mathbf {v}} \in {{[H_0^1(\Omega )]}^2},\forall q \in {M_h},\] where ${M_h}$ is the finite element space associated with the pressure. Several examples for which this operator can be constructed locally illustrate the theory.References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
- 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
- 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
- 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 F. Brezzi & M. Fortin, book in preparation.
- F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, Efficient solutions of elliptic systems (Kiel, 1984) Notes Numer. Fluid Mech., vol. 10, Friedr. Vieweg, Braunschweig, 1984, pp. 11–19. MR 804083
- Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308–340 (Italian). MR 138894
- 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
- Manfred Dobrowolski and Rolf Rannacher, Finite element methods for nonlinear elliptic systems of second order, Math. Nachr. 94 (1980), 155–172. MR 582526, DOI 10.1002/mana.19800940112
- Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI 10.1090/S0025-5718-1980-0559195-7
- 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
- J. Frehse and R. Rannacher, Eine $L^{1}$-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 92–114 (German, with English summary). MR 0471370
- 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 J. A. Nitsche, "Schauder estimates for finite element approximations of second order elliptic boundary value problems," Proc. of the Special Year in Numerical Analysis (I. Babuška, T.-P. Liu and J. Osborn, eds.), Lecture Notes 20, Univ. of Maryland, 1981, pp. 290-343.
- Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
- R. Scholz, Optimal $L_{\infty }$-estimates for a mixed finite element method for second order elliptic and parabolic problems, Calcolo 20 (1983), no. 3, 355–377 (1984). MR 761790, DOI 10.1007/BF02576470
- Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp. 51 (1988), 491-506
- MSC: Primary 65N30; Secondary 65N15, 76-08, 76D99
- DOI: https://doi.org/10.1090/S0025-5718-1988-0935076-9
- MathSciNet review: 935076