Nonlinear Galerkin Method in the Finite Difference Case and Wavelet-like Incremental Unknowns

Abstract

The IMG algorithm (Inertial Manifold-Multigrid algorithm) which uses the first-order incremental unknowns was introduced in Temam and Chen (1991). The algorithm is aimed at numerically implementing inertial manifolds when finite difference discretizations are used. For that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components; the (first-order) Incremental Unknowns (IU) were proposed as a means to realize this decomposition. Our aim in the present article is to propose and study other forms of incremental unknowns, in particular the Wavelet-like Incremental Unknowns (WIU), so-called because of their oscillatory nature.

In this report, we first extend the general convergence results in Temam and Chen (1991) by proving them under slightly weaker conditions. We then present three sets of incremental unknowns (i.e. the first-order as in Temam and Chen (1991), the second-order and wavelet-like incremental unknowns). We show that these incremental unknowns can be used to construct convergent IMG algorithms. Special stress is put on the wavelet-like incremental unknowns since this set of unknowns has the $L^2$ orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms.