Flux pinning and boundary nucleation of vorticity in a mean field model of superconducting vortices

We study a one-dimensional mean field model of superconducting vortices with a finite London penetration depth, flux pinning and nucleation of vorticity at inflow boundary sections. The existence of a unique weak solution is proved and the long time behaviour is studied. A numerical discretization of the model is derived and it is shown that as the time step and the mesh size tend to zero, the discrete solution converges to the unique weak solution of the continuous model. Some numerical computations are presented which illustrate the effects of flux pinning and the finite penetration depth.