Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of elliptic and parabolic problems
Abstract
We present unified frameworks for goal-oriented estimates for elliptic and parabolic problems that combine the dual-weighted residual method with equilibrated flux reconstruction. These frameworks allow to analyze simultaneously different approximation schemes for the space discretization of the primal and the dual problems, as conforming or nonconforming finite element method, discontinuous Galerkin methods, or finite volume method. Our main contribution is the splitting of the error on the quantity of interest into a fully computable estimator and a remainder that is bounded, up to an explicit constant, by the product of the fully computable estimators of the primal and dual problems. Some illustrative numerical examples that validate our theoretical results are finally presented.
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