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

Abstract

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 κ. A comparison is given between the weighted and unweighted Sobolev gradients in a finite element setting. It is seen that for small values of κ, 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.