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

Published online by Cambridge University Press:  15 April 2002

Vivette Girault  and
Jacques-Louis Lions
Vivette Girault
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.
Jacques-Louis Lions
Affiliation:
Collège de France, 75231 Paris Cedex 05, France.
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Abstract

We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size H. In the second step, the problem is linearized by substituting into the non-linear term, the velocity uH computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size h. This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of uH to the error analysis is measured in the L2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H2. Hence, an error of the order of h can be recovered at the second step, provided h = H2.

Keywords

Two gridsa priori estimatesduality.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 5 , September 2001 , pp. 945 - 980
Copyright
© EDP Sciences, SMAI, 2001

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References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Ait Ou Amni, A. and Marion, M., Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. Numer. Math. 62 (1994) 189-213.
Arnold, D., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. CrossRef
Babuska, I., The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192. CrossRef
S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, in Texts in Applied Mathematics 15, Springer-Verlag, New York (1994).
F. Brezzi, On the existence, uniqueness and approximation of saddle-points problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129-151.
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Chorin, A.J., Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955).
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.
Dauge, M., Stationary Stokes and Navier-Stokes systems on two or three-dimensional domains with corners. SIAM J. Math. Anal. 20 (1989) 74-97. CrossRef
Dupont, T. and Scott, L.R., Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34 (1980) 441-463. CrossRef
Foias, C., Manley, O. and Temam, R., Modelization of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO Modél. Anal. Numér. 22 (1988) 93-114. CrossRef
Garcia-Archilla, B. and Titi, E., Postprocessing the Galerkin method: the finite-element case. SIAM J. Numer. Anal. 37 (2000) 470-499. CrossRef
Girault, V. and Lions, J.-L., Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. 58 (2001) 25-57.
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).
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).
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.
P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Pitman Monographs and Studies in Mathematics 24, Pitman, Boston (1985).
Heywood, J., The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980) 639-681. CrossRef
Heywood, J. and Rannacher, R., 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. CrossRef
O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow. In Russian (1961). First English translation, Gordon & Breach, Eds., New York (1963).
Layton, W., A two-level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 26 (1993) 33-38. CrossRef
Layton, W. and Lenferink, W., Two-level Picard-defect corrections for the Navier-Stokes equations at high Reynolds number. Appl. Math. Comput. 69 (1995) 263-274.
Layton, W. and Lenferink, W., Multilevel, A mesh independence principle for the Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 17-30. CrossRef
Leray, J., Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl. 12 (1933) 1-82.
Leray, J., Essai sur des mouvements plans d'un liquide visqueux que limitent des parois. J. Math. Pures Appl. 13 (1934) 331-418.
Leray, J., Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193-248. CrossRef
J.-L. Lions, Équations différentielles opérationnelles 111. Springer-Verlag, Berlin, Heidelberg, New York (1961).
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968).
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Fluids. Oxford University Press, Oxford (1996).
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Fluids. Oxford University Press, Oxford (1998).
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.
Marion, M. and Temam, R., Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26 (1989) 1139-1157. CrossRef
Marion, M. and Temam, R., Nonlinear Galerkin methods: the finite element case. Numer. Math. 57 (1990) 1-22. CrossRef
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.
J. Necas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).
A. Niemistö, FE-approximation of unconstrained optimal control like problems. Report No. 70. University of Jyväskylä (1995).
O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989).
Scott, L.R. and Zhang, S., Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. CrossRef
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1979).
Temam, R., Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115-152. CrossRef
Xu, J., A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994) 231-237. CrossRef
Two-grid, J. Xu finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33 (1996) 1759-1777.