A posteriori error estimation for variational problems with uniformly convex functionals

Abstract

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form \[ \inf \limits _{v\in V} \{ F(v)+G(\Lambda v) \}, \] where $F:V\rightarrow \mathbb {R}$ is a convex lower semicontinuous functional, $G: Y\rightarrow \mathbb {R}$ is a uniformly convex functional, $V$ and $Y$ are reflexive Banach spaces, and $\Lambda :V\rightarrow Y$ is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the