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

Girault, Vivette; Lions, Jacques-Louis

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 5, pp. 945-980.

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 ${𝐮}_H$ 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 ${𝐮}_H$ to the error analysis is measured in the $L^2$ norm in space and time, and thus, for the lowest-degree elements, is of the order of $H^2$. Hence, an error of the order of $h$ can be recovered at the second step, provided $h=H^2$.

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[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. | Zbl

[3] D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. | Zbl

[4] I. Babuška, The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192. | EuDML | Zbl

[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). | MR | Zbl

[6] F. Brezzi, On the existence, uniqueness and approximation of saddle-points problems arising from Lagrange multipliers. RAIRO Anal. Numér. (1974) 129-151. | EuDML | Numdam | Zbl

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl

[8] A.J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | Zbl

[9] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR | Zbl

[10] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). | MR | Zbl

[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. | Zbl

[13] T. Dupont and L.R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34 (1980) 441-463. | Zbl

[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. | Numdam | Zbl

[15] B. Garcia-Archilla and E. Titi, Postprocessing the Galerkin method: the finite-element case. SIAM J. Numer. Anal. 37 (2000) 470-499. | Zbl

[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. | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

[21] J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29 (1980) 639-681. | Zbl

[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. | Zbl

[23] O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow. In Russian (1961). First English translation, Gordon & Breach, Eds., New York (1963). | MR | Zbl

[24] W. Layton, A two-level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 26 (1993) 33-38. | Zbl

[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. | Zbl

[26] W. Layton and W. Lenferink, A Multilevel mesh independence principle for the Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 17-30. | Zbl

[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. | Numdam | Zbl

[28] J. Leray, Essai sur des mouvements plans d'un liquide visqueux que limitent des parois. J. Math. Pures Appl. 13 (1934) 331-418. | JFM | Numdam

[29] J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193-248. | JFM

[30] J.-L. Lions, Équations différentielles opérationnelles 111. Springer-Verlag, Berlin, Heidelberg, New York (1961). | MR | Zbl

[31] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | MR | Zbl

[32] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968). | Zbl

[33] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Fluids. Oxford University Press, Oxford (1996). | MR | Zbl

[34] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Fluids. Oxford University Press, Oxford (1998). | MR | Zbl

[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. | Zbl

[36] M. Marion and R. Temam, Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26 (1989) 1139-1157. | Zbl

[37] M. Marion and R. Temam, Nonlinear Galerkin methods: the finite element case. Numer. Math. 57 (1990) 1-22. | Zbl

[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. | Zbl

[39] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR

[40] A. Niemistö, FE-approximation of unconstrained optimal control like problems. Report No. 70. University of Jyväskylä (1995). | Zbl

[41] O. Pironneau, Finite Element Methods for Fluids. Wiley, Chichester (1989). | MR | Zbl

[42] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl

[43] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1979). | MR | Zbl

[44] R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115-152. | Numdam | Zbl

[45] J. Xu, A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994) 231-237. | Zbl

[46] J. Xu, Two-grid finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33 (1996) 1759-1777. | Zbl