Explicit Discrete Solution for Some Optimization Problems and Estimations with Respect to the Exact Solution

We consider two steady-state heat conduction systems called, $S$ and $S_\alpha$, in a multidimensional bounded domain $D$ for the Poisson equation with source energy $g$. In one system, we impose mixed boundary conditions (temperature $b$ on the boundary $\Gamma_1$, heat flux $q$ on $\Gamma_2$ and an adiabatic condition on $\Gamma_3$). In the other system, the condition on $\Gamma_1$ is replaced by a convective heat flux condition with coefficient $\alpha$. For each of these systems, we consider three associated optimization problems $(P_{i})$ and $(P_{i\alpha })$, $i=1,2,3$, where the variable is the source energy $g$, the heat flux $q$ and the environmental temperature $b$, respectively. In the particular case where $D$ is a rectangle, the explicit continuous optimization variables and the corresponding state of the systems are known. In the present work, by using a finite difference scheme, we obtain the discrete systems $({S^h})$ and ${(S^h_\alpha)}$ and discrete optimization problems ${(P^h_i)}$ and ${(P^h_{i \alpha})}$, $i=1,2,3$, where $h$ is the space step in the discretization. Explicit discrete solutions are found, and convergence and estimation errors results are proved when $h$ goes to zero and when $\alpha$ goes to infinity. Moreover, some numerical simulations are provided in order to test theoretical results. Finally, we note that the use of a three-point finite-difference approximation for the Neumann or Robin boundary condition at the boundary improves the global order of convergence from $O(h)$ to $O(h^2)$.