  We consider a bounded domain $\Omega$ in $\mathbb{R}^{n}$ whose regular
boundary $\partial\Omega$ consists of the union of two disjoint portions
$\Gamma_{1}$ and $\Gamma_{2}$ with $meas(\Gamma_{1})>0$. The convergence of a
family of continuous distributed mixed elliptic optimal control problems
(DMEOCPs) $P_{\alpha}$, governed by elliptic variational equalities (EVE), when
the parameter $\alpha$ goes to infinity was studied in Gariboldi-Tarzia, Appl.
Math. Optim. (2003). It has been proved that the optimal control (OC), and
their corresponding system and adjoint system states (SASSs) are strongly
convergent, in adequate functional spaces, to the OC, and the SASSs of another
CDMEOPC $P$ governed also by an EVE with a different boundary condition on
$\Gamma_{1}$. We consider the discrete approximations $P_{h\alpha}$ and $P_{h}$
of the OCPs $P_{\alpha}$ and $P$ respectively, for each $h>0$ and $\alpha>0$,
through the finite element method with parameter $h$. We also discrete the EVEs
which define the SASSs, and the corresponding cost functional of the DMEOCPs
$P_{\alpha}$ and $P$. The goal is to study the double convergence of this
family of discrete DMEOCPs $P_{h\alpha}$ when $\alpha\to +\infty$ and $h\to 0$
simultaneously. We prove the convergence of the discrete OCs, the discrete
SASSs of the family $P_{h\alpha}$ to the corresponding to the discrete DMEOCP
$P_{h}$ when $\alpha\to +\infty$, for each $h>0$,. We study the convergence of
the discrete OCPs $P_{h\alpha}$ and $P_{h}$ when $h\to 0$ obtaining a
commutative diagram which relates the continuous and discrete DMEOCPs
$P_{h\alpha}$, $P_{h}$, $P_{\alpha}$ and $P$ by taking the limits $h\to 0$ and
$\alpha\to +\infty$ respectively. We also study the double convergence of
$P_{h\alpha}$ to $P$ when $(h,\alpha)\to (0,+\infty)$.
