Bregman divergences and error control via convex duality
It provides theoretical error control tools for a general class of nonlinear problems, benefiting numerical analysts working on finite element methods.
This paper uses convex duality and Bregman divergences to derive error estimates for nonlinear variational problems, including a local efficiency bound for the duality gap estimator, a guaranteed a posteriori bound for non-conforming fields, and a quasioptimal estimate for a Crouzeix-Raviart discretization of the φ-Laplace problem.
Convex duality relations are a useful tool for deriving error estimates for challenging nonlinear and non-smooth variational problems. Applied at the continuous level they can deliver nonlinear analogues of the Prager-Synge a posteriori error identity, while at the discrete level they allow the derivation of minimal regularity a priori estimates. By leveraging elementary properties of Bregman divergences, we obtain three results on the error control via convex duality for a general class of problems: first, we prove a local efficiency bound for the duality gap error estimator, secondly, we derive a guaranteed a posteriori bound for non-conforming fields, and finally, we prove a minimal-regularity quasioptimal estimate for a Crouzeix-Raviart discretisation of the $φ$-Laplace problem.