Finite element approximation of elliptic control problems with constraints on the gradient (Q998638)

The authors consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state of the form \[ \min_{u\in K}\,J(u)={1\over 2} \int_\Omega|y- y_0|^2+ {\alpha\over 2} \int_\Omega|u|^2 \] \[ \text{s.t. }Au= u\text{ in }\Omega\text{ and }y= 0\text{ on }\delta\Omega. \] For this problem a tailored finite element approximation is presented, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation. Some numerical examples are given.