Nonlinear Galerkin methods: The finite elements case. (English) Zbl 0702.65081

Authors’ summary: 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 element) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchial bases. Besides 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.
Reviewer: P.Laasonen


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
Full Text: DOI EuDML


[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) · Zbl 0511.65079
[2] [B] Braess, D.: The contraction number of a multigrid method for solving the Poisson equation. Numer. Math.37, 387-404 (1981) · Zbl 0461.65078
[3] [C] Ciarlet, P.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[4] [FJKSTi] Foias, C., Jolly, M., Kevrikidis, I., Sell, G., Titi, E.: On the computation of inertial manifolds. Phys. Lett. A131, 433-436 (1988)
[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) · Zbl 0663.76054
[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) · Zbl 0694.35028
[7] [Fxx] Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. Differ. Equations73, 309-353 (1988) · Zbl 0643.58004
[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) · Zbl 0692.35053
[9] [FT] Foias, C., Temam, R.: The algebraic approximation of attractors; the finite dimension case. Physica D32, 163-182 (1988) · Zbl 0671.58024
[10] [M1] Marion, M.: Approximate inertial manifolds for reaction-diffusion equations in high space dimension. J. Dyn. Differ. Equations1, 245-267 (1989) · Zbl 0702.35127
[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) · Zbl 0724.65122
[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) · Zbl 0674.35049
[13] [MT] Marion, M., Temam, R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal.26, 1139-1157 (1989) · Zbl 0683.65083
[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) · Zbl 0592.35013
[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) · Zbl 0691.35019
[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 1989
[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) · Zbl 0638.76035
[18] [T3] Temam, R.: Attractors for the Navier-Stokes equations, localization and approximation. J. Fac. Sci. Tokyo, Sec. IA,36, 629-647 (1989) · Zbl 0698.58040
[19] [T4] Temam, R.: Navier-Stokes equations. North-Holland Publishing Company, 3rd revised edition, 1984
[20] [T5] Temam, R.: Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983 · Zbl 0522.35002
[21] [T6] Temam, R.: Sur l’approximation des équation de Navier-Stokes. C.R. Acad. Sci. Paris, Ser. A262, 219-221 (1966) · Zbl 0173.11902
[22] [Y] Yserentant, H.: On the multi-level spliting of finite element spaces. Numer. Math.49, 379-412 (1986) · Zbl 0608.65065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.