Details
| Original language | English |
|---|---|
| Pages (from-to) | 215-236 |
| Number of pages | 22 |
| Journal | Journal of numerical mathematics |
| Volume | 27 |
| Issue number | 4 |
| Early online date | 1 Aug 2019 |
| Publication status | Published - 18 Dec 2019 |
Abstract
In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton's method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.
Keywords
- balancing iteration and discretization errors, dual-weighted residual, finite elements, multiple goal-oriented a posteriori error estimation, p-Laplace
ASJC Scopus subject areas
- Mathematics(all)
- Computational Mathematics
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In: Journal of numerical mathematics, Vol. 27, No. 4, 18.12.2019, p. 215-236.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Multigoal-oriented error estimates for non-linear problems
AU - Endtmayer, Bernhard
AU - Langer, Ulrich
AU - Wick, Thomas
N1 - This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181
PY - 2019/12/18
Y1 - 2019/12/18
N2 - In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton's method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.
AB - In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton's method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.
KW - balancing iteration and discretization errors
KW - dual-weighted residual
KW - finite elements
KW - multiple goal-oriented a posteriori error estimation
KW - p-Laplace
UR - http://www.scopus.com/inward/record.url?scp=85052700548&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1804.01331
DO - 10.48550/arXiv.1804.01331
M3 - Article
AN - SCOPUS:85052700548
VL - 27
SP - 215
EP - 236
JO - Journal of numerical mathematics
JF - Journal of numerical mathematics
SN - 1570-2820
IS - 4
ER -