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In this paper, we consider a family of simultaneous distributed-boundaryoptimal control problems ($P_{\alpha}$) on the internal energy and the heatflux for a system governed by a mixed elliptic variational equality with aparameter $\alpha >0$ and a simultaneous distributed-boundary optimal controlproblem ($P$) governed also by an elliptic variational equality with aDirichlet boundary condition on the same portion of the boundary. We formulatediscrete approximations $\left(P_{h \alpha}\right)$ and $\left(P_h\right)$ ofthe problems $\left(P_\alpha\right)$ and $(P)$ respectively, for each $h>0$ andfor each $\alpha>0$, through the finite element method with Lagrange'striangles of type 1 with parameter $h$ (the longest side of the triangles). Thegoal of this paper is to study the convergence of this family of discretesimultaneous distributed-boundary mixed elliptic optimal control problems$\left(P_{h \alpha}\right)$ when the parameters $\alpha$ goes to infinity andthe parameter $h$ goes to zero simultaneously. We prove the convergence of theproblems $\left(P_{h \alpha}\right)$ to the problem $\left(P_h\right)$ when$\alpha \rightarrow +\infty$, for each $h>0$. We study the convergence of theproblems $\left(P_{h \alpha}\right)$ and $\left(P_h\right)$, for each $\alpha>0$, when $h \rightarrow 0^+$ obtaining a commutative diagram which relates thecontinuous and discrete optimal control problems $\left(P_{h\alpha}\right),\left(P_\alpha\right),\left(P_h\right)$ and $(P)$ by taking thelimits $h \rightarrow 0^+$ and $\alpha \rightarrow +\infty$ respectively. Wealso study the double convergence of $\left(P_{h \alpha}\right)$ to $(P)$ when$(h, \alpha) \rightarrow(0^+,+\infty)$ which represents the diagonalconvergence in the above commutative diagram.

Numerical analysis of a family of simultaneous distributed-boundary  mixed elliptic optimal control problems and their asymptotic behaviour  through a commutative diagram and error estimates