Goal-oriented error estimation based on equilibrated flux and potential reconstruction for the approximation of elliptic and parabolic problems

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 η 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.