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

Julieta Bollati, Mariela C. Olguin, Domingo A. Tarzia

Abstract

We consider two steady-state heat conduction systems called, $S$ and $S_α$, 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 $Γ_1$, heat flux $q$ on $Γ_2$ and an adiabatic condition on $Γ_3$). In the other system, the condition on $Γ_1$ is replaced by a convective heat flux condition with coefficient $α$. For each of these systems, we consider three associated optimization problems $(P_{i})$ and $(P_{iα})$, $i=1,2,3$, 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 ${(S^h_α)}$ and discrete optimization problems ${(P^h_i)}$ and ${(P^h_{i α})}$, $i=1,2,3$, 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 $α$ 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 $O(h^2)$.

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

Abstract

We consider two steady-state heat conduction systems called, and , in a multidimensional bounded domain for the Poisson equation with source energy . In one system, we impose mixed boundary conditions (temperature on the boundary , heat flux on and an adiabatic condition on ). In the other system, the condition on is replaced by a convective heat flux condition with coefficient . For each of these systems, we consider three associated optimization problems and , , where the variable is the source energy , the heat flux and the environmental temperature , respectively. In the particular case where 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 and and discrete optimization problems and , , where is the space step in the discretization. Explicit discrete solutions are found, and convergence and estimation errors results are proved when goes to zero and when 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 to .
Paper Structure (17 sections, 22 theorems, 136 equations, 14 figures, 1 table)

This paper contains 17 sections, 22 theorems, 136 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Discrete Systems for $(S)$ and $(S_\alpha)$
  3. Distributed Optimization Problem with Variable $g$
  4. Discrete Problem ${(P^h_{1})}$ Associated with $(P_1)$
  5. Discrete Problem ${(P^h_{1\alpha})}$ Associated with $(P_{1\alpha})$
  6. Boundary Optimization Problem with Variable $q$
  7. Discrete Problem ${(P^h_2)}$ Associated with $(P_2)$
  8. Discrete Problem ${(P^h_{2\alpha})}$ Associated with $(P_{2\alpha})$
  9. Boundary Optimization Problem with Variable $b$
  10. Discrete Problem ${(P^h_{3})}$ Associated with $(P_3)$
  11. Discrete Problem ${(P^h_{3\alpha})}$ Associated with $(P_{3\alpha})$
  12. Numerical Results
  13. Control Variable $g$
  14. Control Variable $q$
  15. Control Variable $b$
  16. ...and 2 more sections

Key Result

Lemma 1

$$

Figures (14)

  • Figure 1: State of systems $(S)$, $({S^h})$, $(S_\alpha)$ and $({S^h_\alpha})$ using $q=12$, $b=30$, $z_d=40$ and $g=10$.
  • Figure 2: Plot of $u$ and ${u^h_\alpha}$ against $n=1/h$ for different values of $(h,\alpha)$.
  • Figure 3: Plot of $J_1$ and $J^h_1$ against $g$.
  • Figure 4: Plot of $J_{1\alpha}$ and $J^h_{1\alpha}$ for $\alpha=50$ against $g$.
  • Figure 5: Plot of $J_1$ and $J^h_{1\alpha}$ against $g$.
  • ...and 9 more figures

Theorems & Definitions (56)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 46 more