Mathematics of Computation

A posteriori error estimation for variational problems with uniformly convex functionals

Author(s): Sergey I. Repin.
Journal: Math. Comp. 69 (2000), 481-500.
MSC (1991): Primary 65N30
Posted: August 26, 1999
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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,

and

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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Sergey I. Repin
Affiliation: Department of Applied Mathematics St. Petersburg State Technical University 195251, St. Petersburg, Russia
Email: repin@mat.amd.stu.neva.ru

DOI: 10.1090/S0025-5718-99-01190-4
PII: S 0025-5718(99)01190-4
Keywords: A posteriori error estimates, duality theory, nonlinear variational problems
Received by editor(s): April 1, 1997
Posted: August 26, 1999
Additional Notes: This research was supported by INTAS Grant N 96-0835.
Copyright of article: Copyright 2000, American Mathematical Society

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