  We discuss several optimization procedures to solve finite element
approximations of linear-quadratic Dirichlet optimal control problems governed
by an elliptic partial differential equation posed on a 2D or 3D Lipschitz
domain. The control is discretized explicitely using continuous piecewise
linear approximations. Unconstrained, control-constrained, state-constrained
and control-and-state constrained problems are analyzed. A preconditioned
conjugate method for a reduced problem in the control variable is proposed to
solve the unconstrained problem, whereas semismooth Newton methods are
discussed for the solution of constrained problems. State constraints are
treated via a Moreau-Yosida penalization. Convergence is studied for both the
continuous problems and the finite dimensional approximations. In the finite
dimensional case, we are able to show convergence of the optimization
procedures even in the absence of Tikhonov regularization parameter.
Computational aspects are also treated and several numerical examples are
included to illustrate the theoretical results.
