A nonconforming finite element method for an elliptic optimal control problem with constraint on the gradient

This article is concerned with the nonconforming finite element method for distributed elliptic optimal control problems with pointwise constraints on the control and gradient of the state variable. We reduce the minimization problem into a pure state constraint minimization problem. In this case, the solution of the minimization problem can be characterized as fourth-order elliptic variational inequalities of the first kind. To discretize the control problem we have used the bubble enriched Morley finite element method. To ensure the existence of the solution to discrete problems three bubble functions corresponding to the mean of the edge are added to the discrete space. We derive the error in the state variable in $H^2$-type energy norm. Numerical results are presented to illustrate our analytical findings.