A Multigrid-Lanczos Algorithm for the Numerical Solutions of Nonlinear Eigenvalue Problems

We study numerical methods for solving nonlinear elliptic eigenvalue problems which contain folds and bifurcation points. First we present some convergence theory for the MINRES, a variant of the Lanczos method. A multigrid-Lanczos method is then proposed for tracking solution branches of associated discrete problems and detecting singular points along solution branches. The proposed algorithm has the advantage of being robust and can be easily implemented. It can be regarded as a generalization and an improvement of the continuation-Lanczos algorithm. Our numerical results show the efficiency of this algorithm.