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An absolutely stabilized finite element method for the Stokes problem. (English) Zbl 0669.76051
An absolutely stabilized finite element formulation for the Stokes problem is presented in this paper. This new formulation, which is nonsymmetric but stable without employing of any stability constant, can be regarded as a modification of the formulation porposed recently by Th. J. R. Hughes and L. P. Franca [Comput. Methods Appl. Mech. Eng. 65, 85-96 (1987; Zbl 0635.76067)]. Optimal error estimates in \(L^ 2\)-norm for the new stabilized finite element approximation of both the velocity and the pressure fields are established, as well as one in \(H^ 1\)-norm for the velocity field.

MSC:
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179 – 192. · Zbl 0258.65108
[2] 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 loose French summary). · Zbl 0338.90047
[3] Franco Brezzi and Jim Douglas Jr., Stabilized mixed methods for the Stokes problem, Numer. Math. 53 (1988), no. 1-2, 225 – 235. · Zbl 0669.76052
[4] 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.
[5] 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). · Zbl 0467.65062
[6] 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
[7] Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305 – 328. , https://doi.org/10.1016/0045-7825(86)90152-0 T. J. R. Hughes and M. Mallet, Errata: ”A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems”, Comput. Methods Appl. Mech. Engrg. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 329 – 336. , https://doi.org/10.1016/0045-7825(86)90153-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Marc Balestra, 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), no. 1, 85 – 99. , https://doi.org/10.1016/0045-7825(86)90025-3 T. J. R. Hughes, L. P. Franca, and M. Balestra, Errata: ”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. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Michel Mallet, A new finite element formulation for computational fluid dynamics. VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 1, 97 – 112. , https://doi.org/10.1016/0045-7825(87)90125-3 Thomas J. R. Hughes and Leopoldo P. Franca, 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), no. 1, 85 – 96. · Zbl 0635.76067
[8] Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305 – 328. , https://doi.org/10.1016/0045-7825(86)90152-0 T. J. R. Hughes and M. Mallet, Errata: ”A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems”, Comput. Methods Appl. Mech. Engrg. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 329 – 336. , https://doi.org/10.1016/0045-7825(86)90153-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Marc Balestra, 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), no. 1, 85 – 99. , https://doi.org/10.1016/0045-7825(86)90025-3 T. J. R. Hughes, L. P. Franca, and M. Balestra, Errata: ”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. 62 (1987), no. 1, 111. , https://doi.org/10.1016/0045-7825(87)90092-2 Thomas J. R. Hughes, Leopoldo P. Franca, and Michel Mallet, A new finite element formulation for computational fluid dynamics. VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 63 (1987), no. 1, 97 – 112. , https://doi.org/10.1016/0045-7825(87)90125-3 Thomas J. R. Hughes and Leopoldo P. Franca, 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), no. 1, 85 – 96. · Zbl 0635.76067
[9] 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. · Zbl 0317.35037
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