Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints

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.