An analysis technique for stabilized finite element solution of incompressible flows
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 1, pp. 57-89.

This paper presents an extension to stabilized methods of the standard technique for the numerical analysis of mixed methods. We prove that the stability of stabilized methods follows from an underlying discrete inf-sup condition, plus a uniform separation property between bubble and velocity finite element spaces. We apply the technique introduced to prove the stability of stabilized spectral element methods so as stabilized solution of the primitive equations of the ocean.

Classification : 65N30, 76M10
Mots clés : Oseen equations, finite elements, mixed methods, stabilized methods, discrete inf-sup condition, spectral methods, primitive equations
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Rebollo, Tomás Chacón. An analysis technique for stabilized finite element solution of incompressible flows. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 1, pp. 57-89. http://archive.numdam.org/item/M2AN_2001__35_1_57_0/

[1] C. Amrouche and V. Girault, Decomposition of Vector spaces and application to the Stokes problem in arbitrary dimensions. Czeschoslovak Math. J. 44 (1994) 109-140. | Zbl

[2] O. Besson and M. R. Laydi, Some estimates for the anisotropic Navier- Stokes equations and for the hydrostatic approximation. RAIRO-Modél. Math. Anal. Numér. 26 (1992) 855-865. | Numdam | Zbl

[3] I. Babuška, The Finite Element Method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192. | Zbl

[4] C. Baiocchi, F. Brezzi and L. P. Franca, Virtual Bubbles and Galerkin-least-squares type methods (Ga.L.S.). Comput. Methods Appl. Mech. Engrg. 105 (1993) 125-141. | Zbl

[5] C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer-Verlag, Berlin (1992). | MR | Zbl

[6] H. Brézis, Analyse Fonctionnelle. Masson, Paris (1983). | MR | Zbl

[7] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange Multipliers. RAIRO-Anal. Numér. R2 (1974) 129-151. | Numdam | Zbl

[8] F. Brezzi and J. Douglas, Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225-236. | Zbl

[9] F. Brezzi and J. Pitkäranta, On the stabilization of Finite Element approximations of the Stokes problem, in Efficient Solutions for Elliptic Systems. Notes on Numerical Fluid Mechanics 10, W. Hackbusch Ed., Springer-Verlag, Berlin (1984) 11-19. | Zbl

[10] T. Chacón Rebollo, A term by term Stabilization Algorithm for Finite Element solution of incompressible flow problems. Numer. Math. 79 (1998) 283-319. | Zbl

[11] T. Chacón Rebollo and A. Domínguez Delgado, A unified analysis of Mixed and Stabilized Finite Element Solutions of Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 182 (2000) 301-331. | Zbl

[12] T. Chacón Rebollo and F. Guillén González, An intrinsic analysis of existence of solutions for the hydrostatic approximation of Navier-Stokes equations. C. R. Acad. Sci. Paris, Série I 330 (2000) 841-846. | Zbl

[13] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[14] L.P. Franca and S.L. Frey, Stabilized Finite Elements: II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233. | Zbl

[15] L.P. Franca and R. Stenberg, Error analysis fo some Galerkin-Least-Squares methods for the elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680-1697. | Zbl

[16] L.P. Franca, T.J.R. Hughes and R. Stenberg, Stabilized Finite Element Methods, in Incompressible Computational Fluid Dynamics, M.D. Gunzburger and R.A. Nicolaides Eds., Cambridge Univ. Press, New York (1993).

[17] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes equations. Springer-Verlag, Berlin (1988). | MR | Zbl

[18] R. Dautray and L.L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson, Paris (2000). | Zbl

[19] P. Gervasio and F. Saleri, Stabilized Spectral Element approximation for the Navier-Stokes equations. Numer. Methods Partial Differential Eq. 14 (1988) 115-141. | Zbl

[20] T.J.R. Hughes and L.P. Franca, A new Finite Element formulation for CFD: 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) 85-96. | Zbl

[21] T.J.R. Hughes, L.P. Franca and M. Balestra, A new Finite Element formulation for CFD: V. Circumventing the Brezzi-Babuška condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl Mech. Engrg. 59 (1986) 85-99. | Zbl

[22] P. Knobloch and L. Tobiska, Stabilization methods of Bubble type for the Q 1 -Q 1 -Element applied to the incompressible Navier-Stokes equations. ESAIM: M2AN 34 (2000) 85-107. | Numdam | Zbl

[23] R. Lewandowski, Analyse Mathématique et Océanographie. Masson, Paris (1997).

[24] J.L. Lions, R. Temam and S. Wang, New formulation of the primitive equations of the atmosphere and applications. Nonlinearity 5 (1992) 237-288. | Zbl

[25] R. Pierre, Simple C 0 -approximations for the computation of incompressible flows. Comput. Methods Appl Mech. Engrg. 68 (1989) 205-228. | Zbl

[26] G. Russo, Bubble stabilization of Finite Element Methods fo the linearized incompressible Navier-Stokes equations. Comput. Methods Appl Mech. Engrg. 132 (1996) 335-343. | Zbl

[27] L. Tobishka and R. Verfürth, Analysis of a Streamline Diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 107-127. | Zbl

[28] R. Verfürth, Analysis of some Finite Element solutions for the Stokes Problem. RAIRO-Anal. Numér. 18 (1984) 175-182.