  The numerical analysis of a family of distributed mixed optimal control
problems governed by elliptic variational inequalities (with parameter $\alpha
>0$) is obtained through the finite element method when its parameter
$h\rightarrow 0$. We also obtain the limit of the discrete optimal control and
the associated state system solutions when $\alpha\rightarrow \infty$ (for each
$h>0$) and a commutative diagram for two continuous and two discrete optimal
control and its associated state system solutions is obtained when
$h\rightarrow 0$ and $\alpha\rightarrow \infty$. Moreover, the double
convergence is also obtained when $(h, \alpha)\rightarrow(0, \infty)$.
