Error estimates for the full discretization of a nonlocal parabolic model for type-I superconductors

A vectorial nonlocal linear parabolic problem in terms of the magnetic field for superconductors of type-I is considered. This problem is obtained from the quasi-static Maxwell equations, the two-fluid model of London and London, and the nonlocal representation of the superconductive current by Eringen (space convolution). In this contribution, a linear fully discrete approximation scheme is proposed to solve this problem. The convergence of the scheme is proved and the corresponding error estimates are derived under appropriate assumptions. It is also shown how to improve the error estimates under higher regularity.