  In this paper, we consider a family of simultaneous distributed-boundary
optimal control problems ($P_{\alpha}$) on the internal energy and the heat
flux for a system governed by a mixed elliptic variational equality with a
parameter $\alpha >0$ and a simultaneous distributed-boundary optimal control
problem ($P$) governed also by an elliptic variational equality with a
Dirichlet boundary condition on the same portion of the boundary. We formulate
discrete approximations $\left(P_{h \alpha}\right)$ and $\left(P_h\right)$ of
the problems $\left(P_\alpha\right)$ and $(P)$ respectively, for each $h>0$ and
for each $\alpha>0$, through the finite element method with Lagrange's
triangles of type 1 with parameter $h$ (the longest side of the triangles). The
goal of this paper is to study the convergence of this family of discrete
simultaneous distributed-boundary mixed elliptic optimal control problems
$\left(P_{h \alpha}\right)$ when the parameters $\alpha$ goes to infinity and
the parameter $h$ goes to zero simultaneously. We prove the convergence of the
problems $\left(P_{h \alpha}\right)$ to the problem $\left(P_h\right)$ when
$\alpha \rightarrow +\infty$, for each $h>0$. We study the convergence of the
problems $\left(P_{h \alpha}\right)$ and $\left(P_h\right)$, for each $\alpha
>0$, when $h \rightarrow 0^+$ obtaining a commutative diagram which relates the
continuous and discrete optimal control problems $\left(P_{h
\alpha}\right),\left(P_\alpha\right),\left(P_h\right)$ and $(P)$ by taking the
limits $h \rightarrow 0^+$ and $\alpha \rightarrow +\infty$ respectively. We
also study the double convergence of $\left(P_{h \alpha}\right)$ to $(P)$ when
$(h, \alpha) \rightarrow(0^+,+\infty)$ which represents the diagonal
convergence in the above commutative diagram.
