Convergence to equilibrium for discretized gradient-like systems with analytic features
Nour Eddine Alaa, Morgan Pierre · IMA Journal of Numerical Analysis · 2013
We give general conditions which guarantee that the sequence generated by a descent algorithm converges to an equilibrium point. The convergence result is based on the Łojasiewicz gradient inequality; optimal convergence rates are also derived, as well as a stability result. We show how our results apply to a large variety of standard time discretizations of gradient-like flows. Schemes with variable time step are considered and optimal conditions on the maximal step size are derived. Applications to time and space discretizations of the Allen–Cahn equation, the sine–Gordon equation and a damped wave equation are given.