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 \[ inf v ∈ V { F ( v ) + G ( Λ v ) } , \inf \limits _{v\in V} \{ F(v)+G(\Lambda v) \}, \] where F : V → R F:V\rightarrow \mathbb {R} is a convex lower semicontinuous functional, G : Y → R G: Y\rightarrow \mathbb {R} is a uniformly convex functional, V V and Y Y are reflexive Banach spaces, and Λ : V → Y \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 duality error estimate obtained and, thus, can be justified via the duality theory.