Two-grid finite-element schemes for the transient Navier-Stokes problem

Mathematical Modelling and Numerical Analysis

Volumn 35, Issue 5, 2001, Pages 945-980

  Girault, Vivette ^a   Lions, Jacques Louis ^b  

^a SORBONNE UNIVERSITÉ   (France)

^b Paris Sciences et Lettres   (France)

Author keywords

A priori estimates; Duality; Two grids

Indexed keywords

EID: 0035643774     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2001145     Document Type: Article

References (46)

1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
2. A. Ait Ou Amni and M. Marion, Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. Numer. Math. 62 (1994) 189-213.
3. D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344.
4. I. Babuška, The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192.
5. S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, in Texts in Applied Mathematics 15, Springer-Verlag, New York (1994).
6. F. Brezzi, On the existence, uniqueness and approximation of saddle-points problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129-151.
7. F. Brezzi arid M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
8. A. J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Cornput. 22 (1968) 745-762.
9. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
10. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955).
11. M. Crouzeix, Étude d'une méthode de linéarisation. Résolution numérique des équations de Stokes stationnaires. Application aux équations de Navier-Stokes stationnaires, in Approximation et méthodes itératives de résolution d'inéquations variationnelles et de problèmes non linéaires, in IRIA, Cahier 12, Le Chesnay (1974) 139-244.
12. M. Dauge, Stationary Stokes and Navier-Stokes systems on two or three-dimensional domains with corners. SIAM J. Math. Anal. 20 (1989) 74-97.
13. T. Dupont and L.R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34 (1980) 441-463.
14. C. Foias, O. Manley and R. Temam, Modelization of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO Modél. Anal Numér. 22 (1988) 93-114.
15. B. Garcia-Archilla and E. Titi, Postprocessing the Galerkin method: the finite-element case. SIAM J. Numer. Anal. 37 (2000) 470-499.
16. V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. 58 (2001) 25-57.
17. V. Girault and P.-A. Raviart, Finite Element Methods for the Navier-Stokes Equations, in Lecture Notes in Mathematics 749, Springer-Verlag, Berlin, Heidelberg, New York (1979).
18. V. Girault and P.A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms, in Springer Series in Computational Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York (1986).
19. R. Glowinski, Finite element methods for the numerical simulation of unsteady incompressible viscous flow modeled by the Navier-Stokes equations. To appear in Handbook of Numerical Analysis, P.G. Ciarlet and J.-L. Lions, Eds., Elsevier, Amsterdam.
20. P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24, Pitman, Boston (1985).
21. J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980) 639-681.
22. J. Heywood and R. Rannacher, Finite element approximation of the nonstationnary Navier-Stokes problem. Regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275-311.
23. O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow. In Russian (1961). First English translation, Gordon & Breach, Eds., New York (1963).
24. W. Layton, A two-level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 26 (1993) 33-38.
25. W. Layton and W. Lenferink, Two-level Picard-defect corrections for the Navier-Stokes equations at high Reynolds number. Appl. Math. Comput. 69 (1995) 263-274.
26. W. Layton and W. Lenferink, A Multilevel mesh independence principle for the Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 17-30.
27. J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl. 12 (1933) 1-82.
28. J. Leray, Essai sur des mouvements plans d'un liquide visqueux que limitent des parois. J. Math. Pures Appl. 13 (1934) 331-418.
29. J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193-248.
30. J.-L. Lions, Équations différentielles opérationnelles 111. Springer-Verlag, Berlin, Heidelberg, New York (1961).
31. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
32. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968).
33. P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Fluids. Oxford University Press, Oxford (1996).
34. P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Fluids. Oxford University Press, Oxford (1998).
35. P.-L. Lions, On some challenging problems in nonlinear partial differential equations, in Mathematics: Frontiers and Perspectives; Amer. Math. Soc., Providence, RI (2000) 121-135.
36. M. Marion and R. Temam, Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26 (1989) 1139-1157.
37. M. Marion and R. Temam, Nonlinear Galerkin methods: the finite element case. Numer. Math. 57 (1990) 1-22.
38. M. Marion and R. Temam, Navier-Stokes equations: theory and approximation, in Handbook of Numerical Analysis. Vol. VI, P.G. Ciarlet and J.-L. Lions, Eds., Elsevier, Amsterdam (1998) 503-688.
39. J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).
40. A. Niemistö, FE-approximation of unconstrained optimal control like problems. Report No. 70. University of Jyväskylä (1995).
41. O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989).
42. L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493.
43. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1979).
44. R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115-152.
45. J. Xu, A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994) 231-237.
46. J. Xu, Two-grid finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33 (1996) 1759-1777.