Sweeping Algorithms for Inverting the Discrete Ginzburg-Landau Operator

The Ginzburg-Landau equations we study arise in the modeling of superconductivity. One widely used method of discretizing the equations together with the associated periodic boundary conditions, in the case of a rectangle of dimension 2, leads to a ve-point stencil. Solving the system means inverting a sparse matrix of dimension N 2 , where N is the number of grid points on each side of the rectangle. We propose a method that is similar to the shooting technique in the numerical solution of ordinary diierential equations. For small N, the method requires inverting a full matrix of dimension 2N. When N is large, an iterative procedure combining partial sweeping and the technique of divide and conquer (domain decomposition) is appropriate.