Volume 30, Issue 2
Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints

M. Hintermüller, Michael HinzeRonald H.W. Hoppe

J. Comp. Math., 30 (2012), pp. 101-123.

Published online: 2012-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the $\mathcal{ℓ}^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.

  • Keywords

Adaptive finite element method, A posteriori errors, Dualization, Low regularity, Pointwise gradient constraints, State constraints, Weak solutions.

  • AMS Subject Headings

65N30, 90C46, 65N50, 49K20, 49N15, 65K10.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-30-101, author = {Hintermüller , M. and Hinze , Michael and H.W. Hoppe , Ronald}, title = {Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {2}, pages = {101--123}, abstract = {

Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the $\mathcal{ℓ}^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1109-m3522}, url = {http://global-sci.org/intro/article_detail/jcm/8420.html} }
TY - JOUR T1 - Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints AU - Hintermüller , M. AU - Hinze , Michael AU - H.W. Hoppe , Ronald JO - Journal of Computational Mathematics VL - 2 SP - 101 EP - 123 PY - 2012 DA - 2012/04 SN - 30 DO - http://doi.org/10.4208/jcm.1109-m3522 UR - https://global-sci.org/intro/article_detail/jcm/8420.html KW - Adaptive finite element method, A posteriori errors, Dualization, Low regularity, Pointwise gradient constraints, State constraints, Weak solutions. AB -

Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the $\mathcal{ℓ}^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.

M. Hintermüller, Michael Hinze & Ronald H.W. Hoppe. (1970). Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints. Journal of Computational Mathematics. 30 (2). 101-123. doi:10.4208/jcm.1109-m3522
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