Two-side a posteriori error estimates for the dual-weighted residual method

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  • Austrian Academy of Sciences
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OriginalspracheEnglisch
Seiten (von - bis)A371-A394
FachzeitschriftSIAM Journal on Scientific Computing
Jahrgang42
Ausgabenummer1
PublikationsstatusVeröffentlicht - 2020

Abstract

In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency of the error estimator. These results hold true for both nonlinear partial differential equations and nonlinear functionals of interest. Furthermore, the DWR method employed in this work accounts for balancing the discretization error with the nonlinear iteration error. We also perform careful studies of the remainder term that is usually neglected. Based on these theoretical investigations, several algorithms are designed. Our theoretical findings and algorithmic developments are substantiated with some numerical tests. Specifically, we also provide a counterexample in which the saturation assumption is violated.

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Two-side a posteriori error estimates for the dual-weighted residual method. / Endtmayer, B.; Langer, U.; Wick, T.
in: SIAM Journal on Scientific Computing, Jahrgang 42, Nr. 1, 2020, S. A371-A394.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

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note = "Funding Information: ∗Submitted to the journal{\textquoteright}s Methods and Algorithms for Scientific Computing section November 16, 2018; accepted for publication (in revised form) October 18, 2019; published electronically February 19, 2020. https://doi.org/10.1137/18M1227275 Funding: This work was supported by the Austrian Science Fund (FWF) under grant P 29181. The work of the third author was supported by RICAM. †Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria (bernhard.endtmayer@ricam.oeaw.ac.at, ulanger@numa.uni-linz.ac.at). ‡Leibniz Universit{\"a}t Hannover, Institut f{\"u}r Angewandte Mathematik, AG Wissenschaftliches Rechnen, Welfengarten 1, 30167 Hannover, Germany (thomas.wick@ifam.uni-hannover.de).",
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N1 - Funding Information: ∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section November 16, 2018; accepted for publication (in revised form) October 18, 2019; published electronically February 19, 2020. https://doi.org/10.1137/18M1227275 Funding: This work was supported by the Austrian Science Fund (FWF) under grant P 29181. The work of the third author was supported by RICAM. †Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria (bernhard.endtmayer@ricam.oeaw.ac.at, ulanger@numa.uni-linz.ac.at). ‡Leibniz Universität Hannover, Institut für Angewandte Mathematik, AG Wissenschaftliches Rechnen, Welfengarten 1, 30167 Hannover, Germany (thomas.wick@ifam.uni-hannover.de).

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