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 - <p style="text-align: justify;">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.</p>