On the domain geometry dependence of the LBB condition

Mathematical Modelling and Numerical Analysis

Volumn 34, Issue 5, 2000, Pages 935-951

  Chizhonkov, Evgenii V ^a   Olshanskii, Maxim A ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Domain geometry; Eigenvalue problem; Finite elements; Infsup condition; LBB; Stokes problem

Indexed keywords

EID: 0034361360     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2000110     Document Type: Article

References (28)

1. P.P. Aristov and E.V. Chizhonkov, On the Constant in the LBB condition for rectangular domains. Report No. 9535, Dept. of Math. Univ. of Nijmegen, The Netherlands (1995).
2. I. Babuška, The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192.
3. D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa 25 (1997) 131-154.
4. D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 141-158.
5. D. Braess, Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer-Verlag, Berlin, Heidelberg, New York (1997).
6. J.H. Bramble and J.E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximation of elliptic problems. Math. Comp. 50 (1988) 1-17.
7. F. Brezzi, (1974) On the existence, uniqueness and approximation of the saddle-point problems arising from Lagrange multipliers. Numer. Math. 20 (1974) 179-192.
8. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Comp. Math. 15, Springer-Verlag, New York (1991).
9. E.V. Chizhonkov, Application of the Cossera spectrum to the optimization of a method for solving the Stokes Problem. Russ. J. Numer. Anal. Math. Model. 9 (1994) 191-199.
10. M. Crouzeix, Étude d'une méthode de linéarisation. Résolution des équations de Stokes stationaires. Application aux équations des Navier - Stokes stationaires, Cahiers de l'IRIA (1974) 139-244.
11. C.M. Dafermos, Some remarks on Korn's inequality. Z. Angew. Math. Phys. 19 (1968) 913-920.
12. V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin (1986).
13. P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston (1985).
14. M. Gunsburger, Finite element methods for viscous incompressible flows. A guide to the theory, practice and algorithms. Academic Press, London (1989).
15. C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuska-Aziz. Arch. Ration. Mech. Anal. 40 (1971) 384-402.
16. 2. Anal. Math. No. 3 (1977) 177-186.
17. U. Langer and W. Queck, On the convergence factor of Uzawa's algorithm. J. Comp. Appl. Math. 15 (1986) 191-202.
18. S.G. Mikhlin, The spectrum of an operator pencil of the elasticity theory. Uspekhi Mat. Nauk 28 (1973) 43-82; English translation in Russian Math. Surveys, 28.
19. S.G. Mikhlin, The spectrum of an operator pencil of the elasticity theory. Uspekhi Mat. Nauk 28 (1973) 43-82; English translation in Russian Math. Surveys, 28.
20. M.A. Olshanskii, Stokes problem with model boundary conditions. Sbornik: Mathematics 188 (1997) 603-620.
21. M.A. Olshanskii and E.V. Chizhonkov, On the optimal constant in the inf-sup condition for rectangle. Matematicheskie Zametki 67 (2000) 387-396.
22. B.N. Parlett, The Symmetrical Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, New Jersey (1980).
23. R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equation 8 (1992) 97-111.
24. D. Silvester and A. Wathen, Fast iterative solution of stabilized Stokes systems part III: Using block preconditioned. SIAM J. Numer. Anal. 31 (1994) 1352-1367.
25. M. Schäfer and S. Turek, Benchmark computations of laminar flow around cylinder, in Flow Simulation with High-Performance Computers II, E.H. Hirschel Ed., Notes on Numerical Fluid Mechanics, 52, Vieweg (1996) 547-566.
26. G. Strang and G.I. Fix, An analysis of the finite element methods. Prentice-Hall, New-York (1973).
27. S. Turek, Efficient solvers for incompressible flow problems: An algorithmic approach in view of computational aspects. LNCSE 6, Springer, Heidelberg (1999).
28. S. Turek and Chr. Becker, FEATFLOW: Finite element software for the incompressible Navier-Stokes equations: User Manual, Release 1.1. Univ. of Heidelberg (1998) (http://www.featflow.de).