Double convergence of a family of discrete distributed mixed elliptic optimal control problems with a parameter

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 $(h,\alpha)\to (0,+\infty)$.