Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns. (English) Zbl 0798.65093

Earlier, R. Temam [SIAM J. Math. Anal. 21, No. 1, 154-178 (1990; Zbl 0715.35039)] introduced an algorithm for implementing inertial manifolds when finite differences are used. For this the unknown function has to be decomposed into its long and short wavelength components. In this paper the convergence result in the earlier paper is proved under slightly weaker conditions and then three sets of incremental unknowns (IUs) are presented to realize the decomposition. This includes the first order IUs of the previous paper, as well as second order and wavelet like IUs. The latter possess certain orthogonality properties and should be particularly suitable for the approximation of evolution equations by inertial algorithms.


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference 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
35K55 Nonlinear parabolic equations


Zbl 0715.35039
Full Text: DOI EuDML


[1] Atanga, J., Silverster, D. (1991): Preconditionning techniques for the numerical solution of the Stokes problem.Proc. of the Second International Conference on the Application of Super-Computers in Engineering (ASE91), Boston (August, 1991)
[2] Axelsson, O., Barker, V.A. (1984): Finite element solution of boundary value problem: Theory and computation. Academic Press, New York · Zbl 0537.65072
[3] Birkhoff, G., Varga, R.S., Young, D. (1962):Alternating implicit methods. Advances in Computers #3. Academic Press, New York · Zbl 0111.31402
[4] Chen, M., Temam, R. (1991): Incremental unknowns for solving partial differential equations. Numer. Math.59, 255-271 · doi:10.1007/BF01385779
[5] Chen, M., Temam, R. (1993): Incremental unknowns in finite differences: condition number of the Matrix. SIAM J. Matrix Analysis Appl (SIMAX)14 (2) · Zbl 0773.65080
[6] Chen, M., Temam, R. (1993): Incremental unknowns for convection-diffusion equations. Appl. Numer. Math. (to appear) · Zbl 0774.65074
[7] Debussche, A. Marion, M. (1992): On the construction of families of approximate inertial manifolds. J. Differ. Eqn. (to appear) · Zbl 0760.34050
[8] Dryja, M., Widlund, O.B. (1991): Multilevel additive methods of elliptic finite element problems. Preprint · Zbl 0783.65057
[9] Foias, C., Manley, O., Temam, R. (1988)/(1987): Modeling of the interaction small and large eddies in two dimensional turbulent flows. Math. Model. Numer. Anal.22 Sur l’interaction des petits et grands tourbillons dans des ?coulements turbulents. C.R. Acad. Sci. Paris, SerieI 305, 497-500 · Zbl 0624.76072
[10] Foias, C., Sell, G.R., Temam, R. (1988)/(1985): Intertial manifolds for nonlinear evolutionary equations. J. Diff. Eqn.73, 308-353. Vari?t?s inertielles des ?quations diff?rentielles disspatives. C. R. Acad. Sci. Paris, Serie I301, 139-141 · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
[11] Golub, G.H., Meurant, G.A. (1983): R?solution Num?riques des Grands Syst?ms Lin?aires. Collection DER-EDF 49
[12] Golub, G.H., Van Loan, C.F. (1989): Matrix Computations. The John Hopkins University Press, Baltimore
[13] Hale, J. (1988): Asymptotic Behavior of Dissipative systems # 25. Mathematical Surveys and Monograph. AMS, Providence · Zbl 0642.58013
[14] Marion, M. (1989): Approximate inertial manifolds for reaction-diffusion equations in high space dimension. J. Dyn. Differ. Eqn.1, 245-267 · Zbl 0702.35127 · doi:10.1007/BF01053928
[15] Marion, M., Temam, R. (1989): Nonlinear Gakerkin methods. Siam. J. Numer. Anal.26, 1139-1157 · Zbl 0683.65083 · doi:10.1137/0726063
[16] Marion, M., Temam, R. (1990): Nonlinear Galerkin methods: the finite element case. Numer. Math.57, 205-226 · Zbl 0702.65081 · doi:10.1007/BF01386407
[17] Promislow, K., Temam, R. (1991): Approximation and localization of attractors for the Ginzburg-Landau partial differential equations. J. Dyn. Diff. Eqn.3(4), 491-514 · Zbl 0751.34036 · doi:10.1007/BF01049097
[18] Temam, R. (1983): Navier-stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics · Zbl 0522.35002
[19] Temam, R. (1988): Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin Heidelberg New York · Zbl 0662.35001
[20] Temam, R. (1990): Inertial manifolds and multigrid methods. Siam. J. Math. Anal.21, 154-178 · Zbl 0715.35039 · doi:10.1137/0521009
[21] Temam, R. (1991): New emerging methods in numerical analysis: Applications to fluid mechanics. In: M. Gunzberger, N. Nicolaides, eds. Incompressible Computational Fluid Dynamics-Trend and Advances. Cambridge University Press, Cambridge · Zbl 1189.76393
[22] Xu, J. (1989): Theory of multilevel methods. Ph.D. Thesis, Cornell
[23] Xu, J. (1990): Iterative methods by space decomposition and subspace correction: a unifying approach. Penn State Univ., Rep. AM-67
[24] Xu, J. (1992): A new class of iterative methods for nonselfadjoint or indefinite problems. Siam J. Num. Anal.29, 303-319 · Zbl 0756.65050 · doi:10.1137/0729020
[25] Yserentant, H. (1986): On the multi-level spliting of finite element spaces. Numer. Math.49 379-412 · Zbl 0608.65065 · doi:10.1007/BF01389538
[26] Yserentant, H. (1990): Two preconditioners based on the multi-level splitting of finite elements. Numer. Math.58, 163-184 · Zbl 0708.65103 · doi:10.1007/BF01385617
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