Numerische Mathematik

, Volume 57, Issue 1, pp 205–226 | Cite as

Nonlinear Galerkin methods: The finite elements case

  • M. Marion
  • R. Temam


With the increase in the computing power and the advent of supercomputers, the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem. In this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements. The algorithm that we propose, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases. Beside a detailed description of the algorithm, the article includes some technical results on finite elements spaces, and a full study of the stability and convergence of the method.

Subject Classifications

AMS (MOS): 65N30 CR: G1.8 


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  1. [AG] Axelsson, O., Gustafsson, I.: Preconditioning and two-level multigrid methods of arbitrary degree of approximation. Math. Comput.40, n0 161, 219–242 (1983)Google Scholar
  2. [B] Braess, D.: The contraction number of a multigrid method for solving the Poisson equation. Numer. Math.37, 387–404 (1981)Google Scholar
  3. [C] Ciarlet, P.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978Google Scholar
  4. [FJKSTi] Foias, C., Jolly, M., Kevrikidis, I., Sell, G., Titi, E.: On the computation of inertial manifolds. Phys. Lett. A131, 433–436 (1988)Google Scholar
  5. [FMT] Foias, C., Manley, O., Temam, R.: Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. Math. Mod. Numer. Anal.22, 93–114 (1988)Google Scholar
  6. [FNxx] Foias, C., Nicolaenko, B., Sell, G., Temam, R.: Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. J. Math. Pures Appl.67, 197–226 (1988)Google Scholar
  7. [Fxx] Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. Differ. Equations73, 309–353 (1988)Google Scholar
  8. [FSTi] Foias, C., Sell, G., Titi, E.: Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. Dyn. Differ. Equations1, 199–244 (1989)Google Scholar
  9. [FT] Foias, C., Temam, R.: The algebraic approximation of attractors; the finite dimension case. Physica D32, 163–182 (1988)Google Scholar
  10. [M1] Marion, M.: Approximate inertial manifolds for reaction-diffusion equations in high space dimension. J. Dyn. Differ. Equations1, 245–267 (1989)Google Scholar
  11. [M2] Marion, M.: Approximate inertial manifolds for the pattern formation Cahn-Hilliard equation, Proc. Luminy Workshop on Dynamical Systems, in Math. Model. Num. Anal. (M2AN),23, 463–488 (1989)Google Scholar
  12. [MS] Mallet-Paret, J., Sell, G.: Inertial manifolds for reaction diffusion equations in higher space dimensions. J. Am. Math. Soc.1, 805–866 (1988)Google Scholar
  13. [MT] Marion, M., Temam, R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal.26, 1139–1157 (1989)Google Scholar
  14. [NST1] Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors. Physica D16, 155–183 (1985)Google Scholar
  15. [NST2] Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of a class of pattern formation equations, Commun. Partial Differ. Equations14, 245–297 (1989)Google Scholar
  16. [T1] Temam, R.: Dynamical systems, turbulence and numerical solution of the Navier-Stokes equations. In: Proceedings of the 11th International Conference on Numerical Methods in Fluid Dynamics, Dwoyer, D.L., Voigt, R. (Eds.) Lecture Notes in Physics. Berlin-Heidelberg-New York: Springer 1989Google Scholar
  17. [T2] Temam, R.: Variétés inertielles approximatives pour les équations de Navier-Stokes bidimensionnelles. C.R. Acad. Sci. Paris, Ser. II,306, 399–402 (1988)Google Scholar
  18. [T3] Temam, R.: Attractors for the Navier-Stokes equations, localization and approximation. J. Fac. Sci. Tokyo, Sec. IA,36, 629–647 (1989)Google Scholar
  19. [T4] Temam, R.: Navier-Stokes equations. North-Holland Publishing Company, 3rd revised edition, 1984Google Scholar
  20. [T5] Temam, R.: Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983Google Scholar
  21. [T6] Temam, R.: Sur l'approximation des équation de Navier-Stokes. C.R. Acad. Sci. Paris, Ser. A262, 219–221 (1966)Google Scholar
  22. [Y] Yserentant, H.: On the multi-level spliting of finite element spaces. Numer. Math.49, 379–412 (1986)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • M. Marion
    • 1
  • R. Temam
    • 1
  1. 1.Laboratoire d'Analyse NumériqueUniversité Paris-SudOrsayFrance

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