Rebollo, Tomás Chacón
An analysis technique for stabilized finite element solution of incompressible flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 1 , p. 57-89
Zbl 0990.76039 | MR 1811981
URL stable : http://www.numdam.org/item?id=M2AN_2001__35_1_57_0

Classification:  65N30,  76M10
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.

Bibliographie

[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 0823.35140

[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 0765.76017

[3] I. Babuška, The Finite Element Method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192. Zbl 0258.65108

[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 0772.76033

[5] C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer-Verlag, Berlin (1992). MR 1208043 | Zbl 0773.47032

[6] H. Brézis, Analyse Fonctionnelle. Masson, Paris (1983). MR 697382 | Zbl 0511.46001

[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 0338.90047

[8] F. Brezzi and J. Douglas, Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225-236. Zbl 0669.76052

[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 0552.76002

[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 0910.76033

[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 0977.76052

[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 0959.35134

[13] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). MR 520174 | Zbl 0383.65058

[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 0765.76048

[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 0759.73055

[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 851383 | Zbl 0585.65077

[18] R. Dautray and L.L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson, Paris (2000). Zbl 0642.35001

[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 0899.76295

[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 0635.76067

[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 0622.76077

[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 0984.76047

[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 0746.76019

[25] R. Pierre, Simple C 0 -approximations for the computation of incompressible flows. Comput. Methods Appl Mech. Engrg. 68 (1989) 205-228. Zbl 0628.76040

[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 0887.76038

[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 0843.76052

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