The Discrete Brezis-Ekeland Principle

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.