Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints

Abstract

Abstract. A three-dimensional H(curl)-elliptic optimal control problem with distributed control and pointwise constraints on the control is considered. We present a residual-type a posteriori error analysis with respect to a curl-conforming edge element approximation of the optimal control problem. Here, the lowest order edge elements of Nédélec's first family are used for the discretization of the state and the control with respect to an adaptively generated family of simplicial triangulations of the computational domain. In particular, the a posteriori error estimator consists of element and face residuals associated with the state equation and the adjoint state equation. The main results are the reliability of the estimator and its efficiency up to oscillations in terms of the data of the problem. In the last part of the paper, numerical results are included which illustrate the performance of the adaptive approach.Key words. Optimal control of PDEs, H(curl)-elliptic problems, curl-conforming edge elements, residual a posteriori error estimator, reliability and efficiency.AMS subject classifications. Primary: 65K10; Secondary: 49M05,65N30,65N50,78M101. Introduction. This paper is devoted to an a posteriori error analysis of adaptive edge element methods for control constrained distributed optimal control of H(curl)-elliptic problems in R 3 . Adaptive edge element methods for H(curl)-elliptic boundary value problems on the basis of residual-type a posteriori error estimators have been initiated in [7,8,31] and later on considered in [11,35]. A convergence analysis has been provided in [24]. For nonstandard discretizations such as Discontinuous Galerkin methods, we refer to [12,25]. In case of the time-harmonic Maxwell equations, convergence and quasi-optimality of adaptive edge element approximations have been established in [42,46] in the spirit of the results obtained in [13] for linear second order elliptic boundary value problems.

Edge Element Method for Optimal Control of Stationary Maxwell System with Gauss Law

“…In what follows, we shall denote by u ∈ U the unique optimal solution of (P) with the corresponding optimal electric field E ∈ V and the adjoint state p ∈ V ∩ U satisfying (11). Thanks to (11c), we can see that the optimal control enjoys the regularity property (15) u ∈ V ∩ U .…”

“…Example 1 with a smooth optimal solution. As the first example, we consider the model optimal control problem (P) that has an analytical and smooth optimal solution, with the computational domain Ω = (0, 1) 3 , the parameters µ = = 1, ρ = 0, and κ = 1, and the desired state E d given by satisfy the sufficient and necessary optimality system (11). Thus, the optimal solution of (P) is given by u.…”

“…Several mathematical and numerical studies on electromagnetic optimal control problems can be found in the literature. However, they were mainly focused on the cases where the stationary system (1) was replaced by the corresponding timedependent system [2,15,20,21,29] or the curl-curl-elliptic system [11,26,27]; namely, either a time derivative term ∂E/∂t or a zero-order term is added in the first equation of (1). In these cases, the divergence law, i.e., the second equation in (1), can be automatically ensured at the discrete level from the first equation of (1) when edge element methods are used for discretization.…”

A posteriori error analysis for the optimal control of magneto-static fields

“…To the best of the authors' knowledge, this paper presents original contributions on the functional a posteriori error analysis for the optimal control of first-order magneto-static equations. We are only aware of the previous contributions [6,29] on the residual a posteriori error analysis for optimal control problems based on the second-order magnetic vector potential formulation. For recent mathematical results in the optimal control of electromagnetic problems, we refer to [8,9,14,15,24,25,[31][32][33].…”

Optimal Control of Non-Smooth Hyperbolic Evolution Maxwell Equations in Type-II Superconductivity

“…To the best of the author's knowledge, this article is the first study on a PDEconstrained optimization problem governed by nonsmooth and nonlinear evolution Maxwell equations. Almost all studies on the optimal control of Maxwell's equations were devoted to the linear case [8,11,22,15,23,24]. So far, the nonlinear case [25] has only been investigated under a stationary (magnetostatic) and smooth assumption.…”