Details
| Original language | English |
|---|---|
| Pages (from-to) | 351-371 |
| Number of pages | 21 |
| Journal | Computational Methods in Applied Mathematics |
| Volume | 21 |
| Issue number | 2 |
| Early online date | 9 Jan 2021 |
| Publication status | Published - 1 Apr 2021 |
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), no. 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.
Keywords
- Dual-weighted residual method, Efficiency and reliability, Incompressible Navier–Stokes equation, Interpolation, P-Laplace, Saturation assumption
ASJC Scopus subject areas
- Mathematics(all)
- Numerical Analysis
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Computational Methods in Applied Mathematics, Vol. 21, No. 2, 01.04.2021, p. 351-371.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Reliability and efficiency of DWR-Type a posteriori error estimates with smart sensitivity weight recovering
AU - Endtmayer, Bernhard
AU - Langer, Ulrich
AU - Wick, Thomas
PY - 2021/4/1
Y1 - 2021/4/1
N2 - 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), no. 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.
AB - 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), no. 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.
KW - Dual-weighted residual method
KW - Efficiency and reliability
KW - Incompressible Navier–Stokes equation
KW - Interpolation
KW - P-Laplace
KW - Saturation assumption
UR - http://www.scopus.com/inward/record.url?scp=85100028423&partnerID=8YFLogxK
U2 - 10.1515/cmam-2020-0036
DO - 10.1515/cmam-2020-0036
M3 - Article
AN - SCOPUS:85100028423
VL - 21
SP - 351
EP - 371
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
SN - 1609-4840
IS - 2
ER -