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