Minimization of the Ginzburg–Landau energy functional by a Sobolev gradient trust-region method

We describe a generalized Levenberg–Marquardt method for computing critical points of the Ginzburg–Landau energy functional which models superconductivity. The algorithm is a blend of a Newton iteration with a Sobolev gradient descent method, and is equivalent to a trust-region method in which the trust-region radius is defined by a Sobolev metric. Numerical test results demonstrate the method to be remarkably effective.