A taxonomy of consistently stabilized finite element methods for the stokes problem

SIAM Journal on Scientific Computing

Volumn 25, Issue 5, 2004, Pages 1585-1607

  Barth, Teri ^a   Bochev, Pavel ^a   Gunzburger, Max ^b   Shadid, John ^a  

^a SANDIA NATIONAL LABORATORIES   (United States)

^b FLORIDA STATE UNIVERSITY   (United States)

Author keywords

Coercive forms; Iterative solvers; Mixed methods; Stabilized finite element methods

Indexed keywords

DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; ERROR ANALYSIS; PROBLEM SOLVING; STABILITY;

COERSIVE FORMS; ITERATIVE SOLVERS; STABILIZED FINITE ELEMENT METHODS;

FINITE ELEMENT METHOD;

EID: 4344681675     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827502407718     Document Type: Article

References (25)

1. I. BABUSKA AND A. AZIZ, Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 1-359.
2. M. BEHR, L. FRANCA, AND T. TEZDUYAR, Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 104 (1993), pp. 31-48.
3. P. BOCHEV AND R. B. LEHOUCQ, On the finite element solution for the pure Neumann problem, SIAM Rev., submitted.
4. P. BOCHEV AND M. GUNZBURGER, An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations, SIAM J. Numer. Anal., to appear.
5. F. BREZZI, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Modél. Math. Anal. Numér., 8 (1974), pp. 129-151.
6. F. BREZZI AND J. DOUGLAS, Stabilized mixed methods for the Stokes problem, Numer. Math., 53 (1988), pp. 225-235.
7. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978; reprinted as Classics Appl. Math. 40, SIAM, Philadelphia, 2002.
8. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978; reprinted as Classics Appl. Math. 40, SIAM, Philadelphia, 2002.
9. J. DOUGLAS AND J. WANG, An absolutely stabilized finite element method for the Stokes problem, Math. Comp., 52 (1989), pp. 495-508.
10. R. FREUND AND N. NACHTIGAL, QMR: A quasi-minimal residual method for non-Hermitian linear systems, Numer. Math., 60 (1991), pp. 315-339.
11. L. FRANCA, S. FREY, AND T. HUGHES, Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Methods Appl. Mech. Engrg., 95 (1992), pp. 253-276.
12. L. P. FRANCA AND R. STENBERG, Error analysis of Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal., 28 (1991), pp. 1680-1697.
13. L. FRANCA, T. J. R. HUGHES, AND R. STENBERG, Stabilized finite element methods, in Incompressible Computational Fluid Dynamics, M. Gunzburger and R. Nicolaides, eds., Cambridge University Press, Cambridge, UK, 1993, pp. 87-107.
14. V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.
15. A. GREENBAUM, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, 1997.
16. M. GUNZBURGER, Finite Element Methods for Viscous Incompressible Flows, Academic Press, Boston, 1989.
17. I. HARARI, Computational Methods for Problems of Acoustics with Particular Reference to Exterior Domains, Ph.D. thesis, Division of Applied Mathematics, Stanford University, Stanford, CA, 1991.
18. R. HORN AND C. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, London, 1991.
19. A. HOUSEHOLDER, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964.
20. T. HUGHES AND L. 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), pp. 85-96.
21. T. HUGHES, L. FRANCA, AND M. BALESTRA, A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-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.
22. K. JANSEN, S. COLLIS, C. WHITING, AND F. SHAKIB, A better consistency for low-order stabilized finite element methods, Comput. Methods Appl. Mech. Engrg., 174 (1999), pp. 153-170.
23. A. KLAWONN, An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), pp. 540-552.
24. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
25. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS, Boston, 1996.