  In this paper, elliptic control problems with pointwise box constraints on
the state is considered, where the corresponding Lagrange multipliers in
general only represent regular Borel measure functions. To tackle this
difficulty, the Lavrentiev regularization is employed to deal with the state
constraints. To numerically discretize the resulted problem, since the weakness
of variational discretization in numerical implementation, full piecewise
linear finite element discretization is employed. Estimation of the error
produced by regularization and discretization is done. The error order of full
discretization is not inferior to that of variational discretization because of
the Lavrentiev-regularization. Taking the discretization error into account,
algorithms of high precision do not make much sense. Utilizing efficient
first-order algorithms to solve discretized problems to moderate accuracy is
sufficient. Then a heterogeneous alternating direction method of multipliers
(hADMM) is proposed. Different from the classical ADMM, our hADMM adopts two
different weighted norms in two subproblems respectively. Additionally, to get
more accurate solution, a two-phase strategy is presented, in which the
primal-dual active set (PDAS) method is used as a postprocessor of the hADMM.
Numerical results not only verify error estimates but also show the efficiency
of the hADMM and the two-phase strategy.
