Abstract
In 2001, the discovery of the intermetallic compound superconductor $MgB_2$ having a critical temperature of 39K stirred up great interest in using a generalization of the Ginzburg-Landau model, namely the two-band time-dependent Ginzburg-Landau (2B-TDGL) equations, to model the phenomena of two-band superconductivity. In this work, various mathematical and numerical aspects of the two-dimensional, isothermal, isotropic 2B-TDGL equations in the presence of a time-dependent applied magnetic field and a time-dependent applied current are investigated. A new gauge is proposed to facilitate the inclusion of a time-dependent current into the model. There are three parts in this work. First, the 2B-TDGL model which includes a time-dependent applied current is derived. Then, assuming sufficient smoothness of the boundary of the domain, the applied magnetic field, and the applied current, the global existence, uniqueness and boundedness of weak solutions of the 2B-TDGL equations are proved. Second, the existence, uniqueness, and stability of finite element approximations of the solutions are shown and error estimates are derived. Third, numerical experiments are presented and compared to some known results which are related to $MgB_2$ or general two-band superconductivity. Some novel behaviors are also identified.
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