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In this paper we propose a new finite element method for solving ellipticoptimal control problems with pointwise state constraints, including thedistributed controls and the Dirichlet or Neumann boundary controls. The mainidea is to use energy space regularizations in the objective functional, whilethe equivalent representations of the energy space norms, i.e., the$H^{-1}(\Omega)$-norm for the distributed control, the $H^{1/2}(\Gamma)$-normfor the Dirichlet control and the $H^{-1/2}(\Gamma)$-norm for the Neumanncontrol, enable us to transform the optimal control problem into an ellipticvariational inequality involving only the state variable. The ellipticvariational inequalities are second order for the three cases, and includeadditional equality constraints for Dirichlet or Neumann boundary controlproblems. Standard $C^0$ finite elements can be used to solve the resultedvariational inequality. We provide preliminary a priori error estimates for thenew algorithm for solving distributed control problems. Extensive numericalexperiments are carried out to validate the accuracy of the new algorithm.

A new finite element method for elliptic optimal control problems with pointwise state constraints in energy spaces