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We describe a Sobolev gradient method for finding minima of the Ginzburg–Landau functional for superconductivity. This method leads to a particularly simple algorithm which avoids consideration of the nonlinear boundary conditions associated with the Ginzburg–Landau equations. 1. Introduction. There is considerable current interest in finding minima of… (More)
Let X be a separable complete metric space. We characterize completely the innnitesimal generators of semigroups of linear transformations in C b (X), the bounded real-valued continuous functions on X, that are induced by strongly continuous semigroups of continuous transformations in X. In order to do this, C b (X) is equipped with a locally convex… (More)
This paper gives a common theoretical treatment for gradient and Newton type methods for general classes of problems. First, for Euler-Lagrange equations Newton's method is characterized as an (asymptotically) optimal variable steepest descent method. Second, Sobolev gradient type minimization is developed for general problems using a continuous Newton… (More)
A common problem in physics and engineering is the calculation of the minima of energy functionals. The theory of Sobolev gradients provides an efficient method for seeking the critical points of such a functional. We apply the method to functionals describing coarse-grained Ginzburg-Landau models commonly used in pattern formation and ordering processes.