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{}}{_}{{}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 A : 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 duality error estimate obtained and, thus, can be justified via the duality theory.