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Journal of Convex Analysis 16 (2009), No. 1, 071--087
Copyright Heldermann Verlag 2009

The Discrete Brezis-Ekeland Principle

Ulisse Stefanelli
IMATI - CNR, v. Ferrata 1, 27100 Pavia, Italy

We discuss a global-in-time variational approach to the time-discretization of gradient flows of convex functionals in Hilbert spaces. In particular, a discrete version of the celebrated Brezis-Ekeland variational principle is considered. The variational principle consists in the minimization of a functional on entire time-discrete trajectories. The latter functional admits a unique minimizer which solves the classical backward Euler scheme. This variational characterization is exploited in order to re-obtain in a variational fashion and partly extend the known convergence analysis for the Euler method. The relation between this variational technique and a posteriori error control and space approximation is also discussed.

Keywords: Gradient flow, Euler method, Brezis-Ekeland principle, convergence, error control.

MSC: 35K55

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