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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.

MSC:
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
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References:
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