Hyperbolic Maxwell Variational Inequalities for Bean's Critical-State Model in Type-II Superconductivity

This paper focuses on the numerical analysis for three-dimensional Bean's critical-state model in type-II superconductivity. We derive hyperbolic mixed variational inequalities of the second kind for the evolution Maxwell equations with Bean's constitutive law between the electric field and the current density. On the basis of the variational inequality in the magnetic induction formulation, a semidiscrete Ritz--Galerkin approximation problem is rigorously analyzed, and a strong convergence result is proven. Thereafter, we propose a concrete realization of the Ritz--Galerkin approximation through a mixed finite element method based on edge elements of Nédélec's first family, Raviart--Thomas face elements, divergence-free Raviart--Thomas face elements, and piecewise constant elements. As a final result, we prove error estimates for the proposed mixed finite element method.