Finite Element Analysis of an Optimal Control Problem in the Coefficients of Time-Harmonic Eddy Current Equations

Abstract

This paper is concerned with an optimal control problem governed by time-harmonic eddy current equations on a Lipschitz polyhedral domain. The controls are given by scalar functions entering in the coefficients of the curl-curl differential operator in the state equation. We present a mathematical analysis of the optimal control problem, including sensitivity analysis, regularity results, existence of an optimal control, and optimality conditions. Based on these results, we study the finite element analysis of the optimal control problem. Here, the state is discretized by the lowest order edge elements of Nedelec’s first family, and the control is discretized by continuous piecewise linear elements. Our main findings are convergence results of the finite element discretization (without a rate).