Spectral approximation of the H 1 gradient flow of a multi - well potential with bending energy ∗

We consider a fully discrete approximation of the H gradient flow of an energy integral where the energy density is given by the sum of a nonnegative multi-well potential term and a bending energy term. The spatial discretization is based on a Fourier spectral method, which is combined with an implicit Euler time discretization. The numerical method is shown to be stable and to exhibit optimal orders of convergence with respect to its spatial and temporal discretization parameters in the `∞(0, T ;H) and `∞(0, T ;L) norms, without any limitations on the size of the time step in terms of the spatial discretization parameter. AMS subject classifications: 65M70, 74N15