Nonlinear diffusion in type-II superconductors

In this paper we study a nonlinear evolution equation _∂t(σ(|E|)E)+∇×∇×E=F${\mathrm{\partial }}_t(\sigma (\left|\mathit{E}\right|)\mathit{E})+\nabla \times \nabla \times \mathit{E}=\mathit{F}$ in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E$\mathit{E}$ in a conductive medium under the influence of a force F$\mathit{F}$. It is an approximation of Bean's 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.