Convergence to equilibrium for discretized gradient-like systems with analytic features (Q2856622)

General conditions that guarantee that the sequence generated by a descent algorithm converges to an equilibrium point are presented. The convergence result is based on the Łojasiewicz gradient inequality. Optimal convergence rates are extracted and a stability result is given. Various standard time discretizations of gradient-like flows are used for the application of the delivered results and schemes with variable time step are considered and optimal conditions on the maximal step size are derived. Applications to time and space discretizations of Allen-Cahn, sine-Gordon and damped wave equations are demonstrated.