Summary
The IMG algorithm (Inertial Manifold-Multigrid algorithm) which uses the first-order incremental unknowns was introduced in [20]. The IMG algorithm is aimed at numerically implementing inertial manifolds (see e.g. [19]) 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; (first-order) Incremental Unknowns (IU) were proposed in [20] 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 [20] by proving them under slightly weaker conditions. We then present three sets of incremental unknowns (i.e. the first-order as in [20], the second-order and wavelet-like incremental unknowns). We show that these incremental unknown can be used to construct convergent IMG algorithms. Special stress is put on the wavelet-like incremental unknowns since this set of unknowns has theL 2 orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms.
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Chen, M., Temam, R. Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns. Numer. Math. 64, 271–294 (1993). https://doi.org/10.1007/BF01388690
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DOI: https://doi.org/10.1007/BF01388690