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We derive efficient and reliable goal-oriented error estimations, and deviseadaptive mesh procedures for the finite element method that are based on thelocalization of a posteriori estimates. In our previous work [SIAM J. Sci.Comput., 42(1), A371--A394, 2020], we showed efficiency and reliability forerror estimators based on enriched finite element spaces. However, the solutionof problems on a enriched finite element space is expensive. In the literature,it is well known that one can use some higher-order interpolation to overcomethis bottleneck. Using a saturation assumption, we extend the proofs ofefficiency and reliability to such higher-order interpolations. The results canbe used to create a new family of algorithms, where one of them is tested onthree numerical examples (Poisson problem, p-Laplace equation, Navier-Stokesbenschmark), and is compared to our previous algorithm.

Reliability and efficiency of DWR-type a posteriori error estimates with smart sensitivity weight recovering