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Reliability and Efficiency of DWR-Type A Posteriori Error Estimates with Smart Sensitivity Weight Recovering

Bernhard Endtmayer ORCID logo, Ulrich Langer ORCID logo and Thomas Wick ORCID logo

Abstract

We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [B. Endtmayer, U. Langer and T. Wick, Two-side a posteriori error estimates for the dual-weighted residual method, SIAM J. Sci. Comput. 42 2020, 1, A371–A394], we showed efficiency and reliability for error estimators based on enriched finite element spaces. However, the solution of problems on an enriched finite element space is expensive. In the literature, it is well known that one can use some higher-order interpolation to overcome this bottleneck. Using a saturation assumption, we extend the proofs of efficiency and reliability to such higher-order interpolations. The results can be used to create a new family of algorithms, where one of them is tested on three numerical examples (Poisson problem, p-Laplace equation, Navier–Stokes benchmark), and is compared to our previous algorithm.

Funding source: Austrian Science Fund

Award Identifier / Grant number: P 29181

Funding statement: This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 “Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems”.

Acknowledgements

The first two authors would like to thank “Institute of Applied Mathematics” from the Leibniz University Hannover for the organization of their visit in January 2020. Furthermore, the authors would like to express their thanks to the anonymous referees for their helpful hints and valuable suggestions.

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Received: 2020-03-20
Revised: 2020-08-31
Accepted: 2020-12-04
Published Online: 2021-01-09
Published in Print: 2021-04-01

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