Fully discrete linear approximation scheme for electric field diffusion in type-II superconductors

Superconductors are attracting physicists thanks to their ability to conduct electric current with virtually zero resistance. Their nonlinear behaviour opens, on the other hand, challenging problems for mathematicians. Our model of the diffusion of electric field in superconductors is based on three pillars: the eddy-current version of Maxwell’s equations, power law model of type-II superconductivity and linear dependence of magnetic induction on magnetic field. This leads to a time-dependent nonlinear degenerate partial differential equation. We propose a linear fully discrete approximation scheme to solve it. We have proven the convergence of the method and derived the error estimates describing the dependence of the error on the discretization parameters. These theoretical results were successfully confronted with numerical experiments.