Solving the Ginzburg-Landau equations by finite-element methods

We consider finite-element methods for the approximation of solutions of the Ginzburg-Landau equations of superconductivity. The methods are based on a discretization of the Euler-Lagrange equations resulting from the minimization of the free-energy functional. The discretization is effected by requiring the approximate solution to be a piecewise polynomial with respect to a grid. The magnetization versus magnetic field curves obtained through the finite-element methods agree well with analogous calculations obtained by other schemes. We demonstrate, both by analyzing the algorithms and through computational experiments, that finite-element methods can be very effective and efficient means for the computational simulation of superconductivity phenomena and therefore could be applied to determine macroscopic properties of inhomogeneous, anisotropic superconductors.