Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

We present here different situations in which the filtering of high or low modes is used either for stabilizing semi-implicit numerical schemes when solving nonlinear parabolic equations, building adapted damping operators case dispersive equation. We consider provided by mutigrid-like techniques as well resulting from operator with monotone symbols. Our approach applies to several discretization and we focus on finite elements differences. Numerical illustrations are given Cahn-Hilliard, Korteweig-de Vries Kuramoto-Sivashinsky equations.