  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
$H^{-1}(\Omega)$-norm for the distributed control, the $H^{1/2}(\Gamma)$-norm
for the Dirichlet control and the $H^{-1/2}(\Gamma)$-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 elements can be used to solve the resulted
variational inequality. We provide preliminary a priori error estimates for the
new algorithm for solving distributed control problems. Extensive numerical
experiments are carried out to validate the accuracy of the new algorithm.
