On the error analysis of stabilized finite element methods for the Stokes problem. (English) Zbl 1331.76073
Summary: For a family of stabilized mixed finite element methods for the Stokes equations a complete a priori and a posteriori error analysis is given.
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 |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Keywords:
stabilized finite element methods; Galerkin least squares methods; Stokes problem; incompressible elasticity; a priori error estimates; a posteriori error estimatesReferences:
| [1] | D. Boffi, F. Brezzi, and M. Fortin, {\it Mixed Finite Element Methods and Applications}, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. · Zbl 1277.65092 |
| [2] | D. Braess, {\it Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory}, 3rd ed., Cambridge University Press, Cambridge, UK, 2007. · Zbl 1180.65146 |
| [3] | F. Brezzi and J. Pitkäranta, {\it On the stabilization of finite element approximations of the Stokes equations}, in Efficient Solutions of Elliptic Systems, Notes Numer. Fluid Mech. 10, Springer, Berlin, 1984, pp. 11-19. · Zbl 0552.76002 |
| [4] | P. Clément, {\it Approximation by finite element functions using local regularization}, RAIRO Anal. Numér., 9 (1975), pp. 77-84. · Zbl 0368.65008 |
| [5] | J. Douglas, Jr., and J. P. Wang, {\it An absolutely stabilized finite element method for the Stokes problem}, Math. Comp., 52 (1989), pp. 495-508. · Zbl 0669.76051 |
| [6] | L. P. Franca and T. J. R. Hughes, {\it Two classes of mixed finite element methods}, Comput. Methods Appl. Mech. Engrg., 69 (1988), pp. 89-129. · Zbl 0629.73053 |
| [7] | L. P. Franca, T. J. R. Hughes, and R. Stenberg, {\it Stabilized finite element methods}, in Incompressible Computational Fluid Dynamics: Trends and Advances, Cambridge University Press, Cambridge, UK, 2008, pp. 87-107. · Zbl 1189.76339 |
| [8] | L. P. Franca, J. Karam Filho, A. F. D. Loula, and R. Stenberg, {\it A convergence analysis of a stabilized method for the Stokes flow}, Mat. Apl. Comput., 10 (1991), pp. 19-26. · Zbl 0737.76044 |
| [9] | L. P. Franca and R. Stenberg, {\it Error analysis of Galerkin least squares methods for the elasticity equations}, SIAM J. Numer. Anal., 28 (1991), pp. 1680-1697. · Zbl 0759.73055 |
| [10] | V. Girault and P.-A. Raviart, {\it Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms}, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986. · Zbl 0585.65077 |
| [11] | T. Gudi, {\it A new error analysis for discontinuous finite element methods for linear elliptic problems}, Math. Comp., 79 (2010), pp. 2169-2189. · Zbl 1201.65198 |
| [12] | A. Hannukainen, R. Stenberg, and M. Vohralík, {\it A unified framework for a posteriori error estimation for the Stokes problem}, Numer. Math., 122 (2012), pp. 725-769. · Zbl 1301.76049 |
| [13] | T. J. R. Hughes and L. P. Franca, {\it A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces}, Comput. Methods Appl. Mech. Engrg., 65 (1987), pp. 85-96. · Zbl 0635.76067 |
| [14] | T. J. R. Hughes, L. P. Franca, and M. Balestra, {\it A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations}, Comput. Methods Appl. Mech. Engrg., 59 (1986), pp. 85-99. · Zbl 0622.76077 |
| [15] | D. Kay and D. Silvester, {\it A posteriori error estimation for stabilized mixed approximations of the Stokes equations}, SIAM J. Sci. Comput., 21 (1999), pp. 1321-1336. · Zbl 0956.65100 |
| [16] | H.-C. Lee and K.-Y. Kim, {\it A posteriori error estimators for stabilized \(P1\) nonconforming approximation of the Stokes problem}, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 2903-2912. · Zbl 1231.76157 |
| [17] | J. Pitkäranta, {\it Boundary subspaces for the finite element method with Lagrange multipliers}, Numer. Math., 33 (1979), pp. 273-289. · Zbl 0422.65062 |
| [18] | L. Song, Y. Hou, and Z. Cai, {\it Recovery-based error estimator for stabilized finite element methods for the Stokes equation}, Comput. Methods Appl. Mech. Engrg., 272 (2014), pp. 1-16. · Zbl 1296.76087 |
| [19] | R. Verfürth, {\it Error estimates for a mixed finite element approximation of the Stokes equations}, RAIRO Anal. Numér., 18 (1984), pp. 175-182. · Zbl 0557.76037 |
| [20] | R. Verfürth, {\it A posteriori error estimators for the Stokes equations}, Numer. Math., 55 (1989), pp. 309-325. · Zbl 0674.65092 |
| [21] | R. Verfürth, {\it A Posteriori Error Estimation Techniques for Finite Element Methods}, Numer. Math. Sci. Comput., Oxford University Press, Oxford, UK, 2013. · Zbl 1279.65127 |
| [22] | J. P. Wang, Y. Wang, and X. Ye, {\it Unified a posteriori error estimator for finite element methods for the Stokes equations}, Int. J. Numer. Anal. Model., 10 (2013), pp. 551-570. · Zbl 1452.76059 |
| [23] | H. Zheng, Y. Hou, and F. Shi, {\it A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow}, SIAM J. Sci. Comput., 32 (2010), pp. 1346-1360. · Zbl 1410.76206 |
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