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

TL;DR

This paper addresses the mathematical analysis and numerical approximation of a control-constrained optimal control problem governed by Maxwell's equations, providing error estimates and an efficient adaptive scheme.

Contribution

It introduces a finite element approximation scheme for Maxwell's control problem, along with convergence analysis, error estimates, and an a posteriori error estimator.

Findings

The proposed scheme converges with optimal order.

The a posteriori error estimator is reliable and efficient.

Numerical tests confirm the effectiveness of the discretization and error estimation.

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 N\'ed\'elec 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.