Fully discrete scheme for Bean's critical-state model with temperature effects in superconductivity
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 (high-temperature) 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 Nedelec'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.