Nonlinear diffusion in type-II superconductors

In this paper we study a nonlinear evolution equation ∂ t ( σ ( | E | ) E ) + ∇ × ∇ × E = F in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E in a conductive medium under the influence of a force F . It is an approximation of Beanʹs critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of div–curl lemma for transient problems. We also derive some error estimates for the approximate solution.