A method is presented for detecting bifurcations in parameterized nonlinear PDEs. The method is developed in the context of pseudo-arclength continuation with a Newton corrector iteration. An iterative method, such as preconditioned bio-conjugate gradients, is used to solve the linear Newton systems. The method is applicable to any type of imbedded discretization, such as that obtained for finite elements with h refinement, h - p refinement, p refinement, or various multigrid methods. In the present study a spectral element discretization is applied with hierarchic polynomial basis functions. Eigenvectors of the Jacobian contribution of the imbedded subspace provide "seeds" for a Lanczos procedure to produce an efficient scheme. Numerical results for representative problems are presented. An important point is that the full space solution iterate is used for the subspace Jacobian evaluation. © 1991.
Barragy, E., & Carey, G. F. (1991). Bifurcation detection using the lanczos method and imbedded subspaces. IMPACT of Computing in Science and Engineering, 3(1), 76–92. https://doi.org/10.1016/0899-8248(91)90016-N