Fully Discrete Scheme for Bean's Critical-state Model with Temperature Effects in Superconductivity

Abstract

This paper examines the fully discrete analysis of a hyperbolic Maxwell-type variational inequality with temperature effects arising from Bean's critical-state model in type-II (hightemperature) superconductivity. Here, temperature dependence is included in the critical current due to its main importance for the realization of superconducting effects, as confirmed through physical measurements. We propose a fully discrete scheme based on the implicit Euler in time and a mixed FEM in space consisting of Nédélec's edge elements for the electric field and piecewise constant elements for the magnetic induction. Under a regularity assumption on the initial data and a specific setting of the initial approximation, we prove that the fully discrete scheme admits a unique solution satisfying the discrete Gauss law, which turns out to be the key point for our analysis. Hereafter, stability estimates for the zero-oder and first-order terms of the proposed approximation are investigated. Our main result is the strong convergence of the fully discrete scheme without any further regularity assumption. In particular, the convergence result yields the existence of a unique solution for the variational inequality obeying the physical Gauss law. The final part of the paper is devoted to the a priori error analysis under an additional Lipschitz assumption on the critical current and the temperature distribution as well as a higher regularity assumption on the continuous solution. We close this paper by presenting some 3D numerical results, which in particular confirm the physical Meissner-Ochsenfeld effect in superconductivity.

Maxwell quasi-variational inequalities in superconductivity

“…This paper is a continuation of the recent papers [32,33] on hyperbolic Maxwell variational inequalities (VI), including those arising in high-temperature superconductivity and electromagnetic shielding (cf. [14,27,28]). The goal of the present paper is to explore hyperbolic Maxwell QVI arising from the Bean-Kim model (B1)-(B3) with magnetic field and temperature dependence in the critical current j c = j c (θ, H).…”

“…also [32] for a more general class of hyperbolic Maxwell VI). The pure temperature dependence j c = j c (x, θ(x, t)) was considered in the subsequent paper [27] focusing on its fully discrete analysis. The present paper extends [27,30]: We consider the more realistic case j c = j c (x, θ(x, t), H(x, t)) with less regularity requirement for the data u, θ, and (E 0 , H 0 ).…”

“…The pure temperature dependence j c = j c (x, θ(x, t)) was considered in the subsequent paper [27] focusing on its fully discrete analysis. The present paper extends [27,30]: We consider the more realistic case j c = j c (x, θ(x, t), H(x, t)) with less regularity requirement for the data u, θ, and (E 0 , H 0 ). Note that, since our problem features a QVI character, the prior developed VI-techniques in [27,30,32] cannot be directly applied to (QVI) and require certain extensions.…”

“…We investigate the stability analysis of the resulting time-discrete magnetic and electric fields, including their difference quotients (Lemmas 4.5 and 4.6). Differently from [27,Assumption 2.2], our stability analysis does not rely on any compatibility condition for the initial data. We circumvent this issue by introducing an auxiliary current density (33) and initial difference quotients (34), that preserve the pivotal QVI structure at the initial time (35).…”

Optimal control of elliptic variational inequalities with bounded and unbounded operators

“…We notice that the PDE-constraint in (58) is motivated by the timediscretization of Bean's critical-state model for type-II superconductivity (cf. [30,31,28,32]). We also refer to [29] for our previous contribution towards the optimal control of nonlinear elliptic Maxwell equations.…”