Goal-oriented error estimation based on equilibrated flux and potential reconstruction for the approximation of elliptic and parabolic problems
Résumé
We present a unified framework for goal-oriented estimates for elliptic and parabolic
problems that combines the dual-weighted residual method with equilibrated flux and potential
reconstruction. These frameworks allow to analyze simultaneously different approximation
schemes for the space discretization of the primal and the dual problems such as conforming
or nonconforming finite element methods, discontinuous Galerkin methods, or the finite
volume method. Our main contribution is twofold: first in a unified framework we prove
the splitting of the error into a fully computable estimator $\eta$ and a remainder, second this
remainder is estimated by the product of the fully computable energy-based error estimators
of the primal and dual problems. Some illustrative numerical examples that validate our
theoretical results are finally presented.
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