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Error estimates for a bilinear optimal control problem of Maxwell's equations

Francisco Fuica, Felipe Lepe, Pablo Venegas

TL;DR

This work addresses a bilinear optimal control problem for time-harmonic Maxwell's equations with a finite-dimensional control entering as a coefficient. It develops a rigorous framework: proving existence and first- and second-order optimality conditions, and constructing a finite element discretization based on the lowest-order Nédélec elements. A comprehensive a priori and a posteriori error theory is established for the state, adjoint, and reduced control problems, including a residual-based estimator that splits into state and adjoint contributions and is proven reliable and locally efficient. Numerical experiments, including smooth, 3D-L-shaped, and discontinuous-parameter scenarios, validate the theory and show the adaptive method achieving favorable convergence and robustness in challenging geometries and data.

Abstract

We consider a control-constrained optimal control problem subject to time-harmonic Maxwell's equations; the control variable belongs to a finite-dimensional set and enters the state equation as a coefficient. We derive existence of optimal solutions, and analyze first- and second-order optimality conditions. We devise an approximation scheme based on the lowest order Nédélec finite elements to approximate optimal solutions. We analyze convergence properties of the proposed scheme and prove a priori error estimates. We also design an a posteriori error estimator that can be decomposed as the sum two contributions related to the discretization of the state and adjoint equations, and prove that the devised error estimator is reliable and locally efficient. We perform numerical tests in order to assess the performance of the devised discretization strategy and the a posteriori error estimator.

Error estimates for a bilinear optimal control problem of Maxwell's equations

TL;DR

This work addresses a bilinear optimal control problem for time-harmonic Maxwell's equations with a finite-dimensional control entering as a coefficient. It develops a rigorous framework: proving existence and first- and second-order optimality conditions, and constructing a finite element discretization based on the lowest-order Nédélec elements. A comprehensive a priori and a posteriori error theory is established for the state, adjoint, and reduced control problems, including a residual-based estimator that splits into state and adjoint contributions and is proven reliable and locally efficient. Numerical experiments, including smooth, 3D-L-shaped, and discontinuous-parameter scenarios, validate the theory and show the adaptive method achieving favorable convergence and robustness in challenging geometries and data.

Abstract

We consider a control-constrained optimal control problem subject to time-harmonic Maxwell's equations; the control variable belongs to a finite-dimensional set and enters the state equation as a coefficient. We derive existence of optimal solutions, and analyze first- and second-order optimality conditions. We devise an approximation scheme based on the lowest order Nédélec finite elements to approximate optimal solutions. We analyze convergence properties of the proposed scheme and prove a priori error estimates. We also design an a posteriori error estimator that can be decomposed as the sum two contributions related to the discretization of the state and adjoint equations, and prove that the devised error estimator is reliable and locally efficient. We perform numerical tests in order to assess the performance of the devised discretization strategy and the a posteriori error estimator.
Paper Structure (25 sections, 24 theorems, 142 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 25 sections, 24 theorems, 142 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation
  4. Piecewise smooth fields
  5. The state equation
  6. The model problem
  7. Finite element approximation
  8. A priori error estimates for the model problem
  9. A posteriori error estimate for the model problem
  10. The optimal control problem
  11. Existence of solutions
  12. Optimality conditions
  13. First-order optimality condition
  14. Second-order optimality conditions
  15. Finite element approximation
  16. ...and 10 more sections

Key Result

Theorem 3.1

\newlabelthm:extra_reg_Maxwell0 Let $\mathbf{y} \in \mathbf{H}_{0}(\mathop{\mathbf{curl}}\nolimits,\Omega)$ be the unique solution to problem eq:weak_eq. Then, $\mathrm{(i)}$ if $\mathbf{f}\in \mathbf{H}(\textnormal{div},\Omega)$ and $\varepsilon_{\sigma},\mu\in P\mathrm W^{1,\infty}(\Omega)$, the

Figures (4)

  • Figure 1: Test 2. Left: Initial mesh for the L-shaped domain. Right: Comparison between error curves for uniform and adaptive refinements, together with computed values of estimator $\mathcal{E}_{ocp,\mathscr{T}_{h}}$.
  • Figure 2: Test 2. Intermediate adaptively refined meshes with $15408$ (left) and $263463$ (right) number of elements using the estimator $\mathcal{E}_{ocp,\mathscr{T}_{h}}$.
  • Figure 3: Test 3. Adaptively refined mesh with 1626796 number of elements and the corresponding cross sections of the mesh.
  • Figure 4: Test 3. Left: Numerical solution $\boldsymbol{y}^{*}_h$ (magnitude and vector field) computed on an adaptively refined mesh with 1626796 number of elements. Right: Comparison between the convergence of the estimators $\mathcal{E}_{st,\mathscr{T}_{h}}$ and $\mathcal{E}_{ad,\mathscr{T}_{h}}$ with uniform and adaptive refinement.

Theorems & Definitions (42)

  • Theorem 3.1: extra regularity
  • Proof 1
  • Theorem 3.2: error estimates
  • Theorem 3.3: global reliability of $\mathcal{E}$
  • Proof 2
  • Remark 4.1: extensions
  • Theorem 4.2: differentiability properties of $\mathcal{S}$
  • Proof 3
  • Corollary 4.3: differentiability properties of $j$
  • Theorem 4.4: existence of optimal solutions
  • ...and 32 more