A finite element analysis of high kappa, high field Ginzburg-Landau type model of superconductivity

This work is concerned with the formulation and analysis of a simplified GinzburgLandau type model of superconductivity which is valid for large K and large magnetic field strengths. This model, referred to as the High kappa model, is derived via formal asymptotic expansion of the full, time-dependent Ginzburg-Landau equations. The model accounts for the effects of both applied magnetic fields and currents of constant magnitude. A notable feature of our model is that the systems for the leading order terms for the magnetic potential and the order parameter are decoupled.

Finite element approximations of the High kappa model are introduced using standard Galerkin discretization in space and Backward-Euler and Crank-Nicolson discretization schemes in time. We establish existence and uniqueness results for the fully-discrete equations as well as optimal L2 and HI error estimates for the Backward-Euler-Galerkin and the Crank-Nicolson-Galerkin problems.

Computational experiments are performed with several combinations of spatial and time discretizations of the High kappa model equations. Among other things our numerical approximations show good agreement for rates of convergence in space and time with the corresponding theoretical values. Finally, some well known steady-state and dynamic phenomena valid for type II superconductors are illustrated numerically.