Numerical approximation of time evolution related to Ginzburg-Landau functionals using weighted Sobolev gradients

Sobolev gradients have been discussed in Sial et al. (2003) as a method for energy minimization related to Ginzburg-Landau functionals. In this article, a weighted Sobolev gradient approach for the time evolution of a Ginzburg-Landau functional is presented for different values of kappa. A comparison is given between the weighted and unweighted Sobolev gradients in a finite element setting. It is seen that for small values of kappa, the weighted Sobolev gradient method becomes more and more efficient compared to using the unweighted Sobolev gradient. A comparison with Newton's method is given where the failure of Newton's method is demonstrated for a test problem. (C) 2013 Elsevier Ltd. All rights reserved.