On finite element approximation of variational inequalities arising from elliptic control problems

Two systems of variational equations and inequalities, which characterize the solution of a distributed elliptic control problem governed by the Laplacian with pointwise control and state constraints respectively, are discretized by a mixed finite element method using piecewise linear shape functions. The resulting error estimates of the approximation of the optimal state are derived in H 1,2, whereas the corresponding error estimates for the optimal control are given in L 2 , if the cost functional is strictly coercive, otherwise in negative norms.