Three remarks on the convergence of some discretized second order gradient-like systems

We study several discretizations of a second order gradient-like system with damping. We first consider an explicit scheme with a linear damping in finite dimension. We prove that every solution converges if the nonlinearity satisfies a global Lojasiewicz inequality. Convergence rates are also established. In the case of a strong nonlinear damping, we prove convergence of every solution for a fully implicit scheme in the one-dimensional case, even if the nonlinearity does not satisfy a Lojasiewicz inequality. The optimality of the damping is also established. Numerical simulations illustrate the theoretical results.