Numerical determination of vortices in superconductors: simulation of cooling

We use a gradient descent method to numerically calculate critical points of the Ginzburg–Landau energy functional for a two-dimensional domain, possibly including holes. By directly minimizing the functional we avoid the difficulty of treating the Ginzburg–Landau equations with their associated nonlinear boundary conditions. The descent method is made efficient by the use of Sobolev gradients. In order to find the minimum-energy critical point we simulate a cooling process in which we compute a sequence of critical points, each associated with a slightly lower temperature. The solution at each temperature value serves as a good initial estimate for the next value. We present test results that demonstrate the effectiveness of the method.