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Convergence to equilibrium for discretized gradient-like systems with analytic features. (English) Zbl 1283.65063

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.

MSC:

65K10 Numerical optimization and variational techniques
35Q35 PDEs in connection with fluid mechanics
35Q40 PDEs in connection with quantum mechanics
49J20 Existence theories for optimal control problems involving partial differential equations
35L70 Second-order nonlinear hyperbolic equations
49M25 Discrete approximations in optimal control
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