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

1. Problem Statement

The paper addresses steady-state heat conduction problems governed by the Poisson equation in a multidimensional bounded domain $\Omega \subset \mathbb{R}^n$ (specifically analyzed in a rectangular domain). The study focuses on two distinct systems:

• System (S): Features mixed boundary conditions: Dirichlet (fixed temperature $b$) on $\Gamma_1$, Neumann (fixed heat flux $q$) on $\Gamma_2$, and adiabatic (zero flux) on $\Gamma_3$.
• System (S$_\alpha$): Similar to (S), but the Dirichlet condition on $\Gamma_1$ is replaced by a Robin boundary condition (convective heat flux) involving a coefficient $\alpha > 0$.

Associated with these systems are three optimal control problems for each system ($P_i$ and $P_{i\alpha}$ for $i=1,2,3$), where the control variables are:

1. $g$: Distributed internal energy source.
2. $q$: Boundary heat flux on $\Gamma_2$.
3. $b$: Environmental temperature on $\Gamma_1$.

The objective is to minimize a quadratic cost functional $J$ that balances the deviation of the state (temperature) from a desired target $z_d$ against the magnitude of the control variable (regularization).

While explicit continuous solutions for these problems on a rectangular domain were previously known, this paper aims to derive explicit discrete solutions using finite difference methods and rigorously analyze their convergence and error estimates.

2. Methodology

The authors employ a Finite Difference Method (FDM) to discretize the continuous systems and optimization problems.

• Discretization:

  • The domain is discretized with a uniform mesh size $h = x_0/n$.
  • Interior Nodes: A classical centered second-order finite difference approximation is used for the second derivative ($\partial^2 u / \partial x^2$).
  • Boundary Conditions (Standard):
    • Neumann conditions are approximated using a first-order backward difference.
    • Robin conditions are approximated using a first-order forward difference.
  • This leads to linear systems ($S_h$ and $S_{h\alpha}$) with tridiagonal coefficient matrices.

• Optimization:

  • The continuous cost functionals are discretized to form discrete cost functionals ($J^h_i$ and $J^h_{i\alpha}$).
  • Since the control variables are assumed constant, the discrete optimization problems reduce to minimizing quadratic functions of a single variable, allowing for the derivation of explicit formulas for the optimal discrete controls ($g^h_{op}, q^h_{op}, b^h_{op}$).

• Convergence Analysis:

  • The paper derives explicit error estimates comparing the discrete solutions ($u^h, g^h_{op}$, etc.) with the exact continuous solutions ($u, g_{op}$, etc.).
  • It analyzes the double convergence behavior: as the mesh size $h \to 0$ and the convective coefficient $\alpha \to \infty$.

• Improved Approximation (Section 7):

  • To address the low order of convergence ($O(h)$) caused by first-order boundary approximations, the authors propose a three-point finite difference scheme for the Neumann and Robin boundary conditions.
  • This modification introduces a "ghost point" formulation, effectively increasing the accuracy of the boundary treatment.

3. Key Contributions

1. Explicit Discrete Solutions: The paper provides closed-form analytical expressions for the discrete state variables and the optimal control variables for all three types of optimization problems (distributed source, flux, and temperature). This is a significant contribution as it avoids iterative numerical solvers for the optimization step.
2. Rigorous Error Estimation: The authors prove that the discrete solutions converge to the continuous ones with first-order accuracy ($O(h)$) in the $L^2$ norm when using standard boundary approximations. They provide explicit constants for these error bounds.
3. Double Convergence Analysis: The study establishes the commutativity of the limits: the discrete solution of the Robin problem converges to the discrete solution of the Dirichlet problem as $\alpha \to \infty$, and both converge to the continuous solution as $h \to 0$.
4. Order Improvement: By utilizing a three-point approximation for the boundary conditions, the authors demonstrate that the global order of convergence for the state variable improves from $O(h)$ to $O(h^2)$, while the derivative error remains $O(h)$.

4. Results

• Convergence Rates:
  • Standard FDM: The error $\|u - u^h\|_H$ behaves as $C_1 h$.
  • Improved FDM (3-point): The error $\|u - \tilde{u}^h\|_H$ behaves as $D_1 h^2$.
• Optimal Controls: The explicit formulas for the optimal controls show that the discrete optimal controls ($g^h_{op}$, etc.) differ from the continuous ones by terms proportional to $h$.
• Numerical Simulations:
  • Simulations on a unit square domain ($x_0=y_0=1$) confirm the theoretical convergence rates.
  • Tables and figures illustrate that halving the mesh size $h$ approximately halves the $L^2$ error, confirming the $O(h)$ behavior.
  • The simulations verify that as $\alpha$ increases, the solutions for the Robin system ($S_\alpha$) converge to those of the Dirichlet system ($S$).

5. Significance

• Benchmarking: The derived explicit discrete solutions serve as rigorous benchmarks for testing the accuracy and reliability of general-purpose numerical methods (like Finite Element Methods) for optimal control problems, where explicit solutions are rarely available.
• Efficiency: The ability to compute optimal controls explicitly (without iteration) significantly reduces computational cost for this class of problems.
• Boundary Treatment Insight: The paper highlights the critical impact of boundary condition discretization on global convergence orders. It demonstrates that even with a second-order interior scheme, a first-order boundary scheme limits the global accuracy to $O(h)$, and that higher-order boundary schemes are necessary to achieve $O(h^2)$ global accuracy.
• Limitations and Future Work: The current analysis is restricted to rectangular domains to facilitate the derivation of explicit solutions. The authors suggest extending this methodology to general domains and coordinate systems (polar, spherical) as future work.

In summary, this paper bridges the gap between continuous optimal control theory and discrete numerical implementation by providing exact discrete formulas and proving their convergence properties, while offering a practical technique to enhance numerical accuracy through improved boundary approximations.