Modelling and Numerical Solutions of a Gauge Periodic Time Dependent Ginzburg-Landau Model for Type-II Superconductors

Abstract

In this paper we seek the solutions of the time dependent Ginzburg-Landau model for type-II superconductors such that the associated physical observables are spatially periodic with respect to some lattice whose basic lattice cell is not necessarily rectangular. After appropriately fixing the gauge, the model can be formulated as a system of nonlinear parabolic partial differential equations with quasi-periodic boundary conditions. We first give some results concerning the existence, uniqueness and regularity of solutions and then we propose a semi-implicit finite element scheme solving the system of nonlinear partial differential equations and show the optimal error estimates both in the $L^2$ and energy norm. We also report on some numerical results at the end of the paper.