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A posteriori error estimation
for variational problems
with uniformly convex functionals

Author: Sergey I. Repin
Journal: Math. Comp. 69 (2000), 481-500
MSC (1991): Primary 65N30
Published electronically: August 26, 1999
MathSciNet review: 1681096
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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

\begin{displaymath}\inf\limits _{v\in V} \{ F(v)+G(\Lambda v) \}, \end{displaymath}

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 duality error estimate obtained and, thus, can be justified via the duality theory.

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Additional Information

Sergey I. Repin
Affiliation: Department of Applied Mathematics St. Petersburg State Technical University 195251, St. Petersburg, Russia

Keywords: A posteriori error estimates, duality theory, nonlinear variational problems
Received by editor(s): April 1, 1997
Published electronically: August 26, 1999
Additional Notes: This research was supported by INTAS Grant N 96-0835.
Article copyright: © Copyright 2000 American Mathematical Society