Details
| Original language | English |
|---|---|
| Pages (from-to) | A371-A394 |
| Journal | SIAM Journal on Scientific Computing |
| Volume | 42 |
| Issue number | 1 |
| Publication status | Published - 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.
Keywords
- Dual-weighted residual method, Efficiency and reliability of the error estimator, Incompressible Navier{Stokes equations, Multiple goal functionals, P-Laplacian, Saturation assumption
ASJC Scopus subject areas
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
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In: SIAM Journal on Scientific Computing, Vol. 42, No. 1, 2020, p. A371-A394.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Two-side a posteriori error estimates for the dual-weighted residual method
AU - Endtmayer, B.
AU - Langer, U.
AU - Wick, T.
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).
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - Dual-weighted residual method
KW - Efficiency and reliability of the error estimator
KW - Incompressible Navier{Stokes equations
KW - Multiple goal functionals
KW - P-Laplacian
KW - Saturation assumption
UR - http://www.scopus.com/inward/record.url?scp=85083762193&partnerID=8YFLogxK
U2 - 10.1137/18m1227275
DO - 10.1137/18m1227275
M3 - Article
AN - SCOPUS:85083762193
VL - 42
SP - A371-A394
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
SN - 1064-8275
IS - 1
ER -