Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints
Year: 2012
Author: M. Hintermüller, Michael Hinze, Ronald H.W. Hoppe
Journal of Computational Mathematics, Vol. 30 (2012), Iss. 2 : pp. 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.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1109-m3522
Journal of Computational Mathematics, Vol. 30 (2012), Iss. 2 : pp. 101–123
Published online: 2012-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Adaptive finite element method A posteriori errors Dualization Low regularity Pointwise gradient constraints State constraints Weak solutions.