In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the \(H^{-1}(\varOmega )\) -norm for the distributed control, the \(H^{1/2}(\varGamma )\) -norm for the Dirichlet control and the \(H^{-1/2}(\varGamma )\) -norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard \(C^0\) finite element methods can be used to solve the resulted variational inequalities. We provide preliminary a priori error estimates for the new method for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new method.
