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

M. Hintermuller, Michael Hinze & Ronald H.W. Hoppe

DOI: 10.4208/jcm.1109-m3522

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

Published online: 2012-04

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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 \ell^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 = {M. Hintermuller, Michael Hinze and Ronald H.W. Hoppe}, title = {Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient STATE-Constratints}, 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 \ell^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-Constratints AU - M. Hintermuller, Michael Hinze & Ronald H.W. Hoppe 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 KW - A posteriori errors KW - Dualization KW - Low regularity KW - Pointwise gradient constraints KW - State constraints KW - Weak solutions AB - Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the \ell^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. Hintermuller, Michael Hinze & Ronald H.W. Hoppe. (1970). Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient STATE-Constratints. Journal of Computational Mathematics. 30 (2). 101-123. doi:10.4208/jcm.1109-m3522
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