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

TL;DR

This paper develops explicit discrete solutions for certain heat conduction optimization problems using finite difference schemes, proving convergence and error estimates, and demonstrating improved accuracy with advanced boundary approximations.

Contribution

It introduces explicit discrete solutions for optimization problems in heat conduction systems and establishes convergence and error bounds as discretization parameters vary.

Findings

Explicit discrete solutions are derived for the optimization problems.

Convergence and error estimates are proved as the grid size decreases.

Using a three-point finite-difference scheme improves convergence order to O(h^2).

Abstract

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 $O(h^2)$.