Multigrid Methods for Locating Singularities in Bifurcation Problems

Abstract

This paper deals with multigrid methods for locating singular points for nonlinear equations, such as limit points and bifurcation points (perfect or imperfect), and is restricted to the self-adjoint case. A minimization problem that defines singular points is formulated. It treats uniformly limit points and bifurcation points unlike other methods that are designed to solve for one of the two. So it is particularly useful when the type of singularity, or even its existence, is not known in advance. Efficient multigrid methods for locating singular points based on the minimization problem are described. They solve the problems to the level of discretization errors in just a few work units (about 10 or fewer), where a work unit is the work involved in one local relaxation on the finest grid.