A posteriori error estimation for variational problems with uniformly convex functionals

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∈V {F (v) +G(Λv)}, where F : V → R is a convex lower semicontinuous functional, G : Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ : V → 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.