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Bifurcation detection using the Lanczos method and imbedded subspaces. (English) Zbl 0753.65085

Summary: A method is presented for detecting bifurcations in parameterized nonlinear partial differential equations. The method is developed in the context of pseudo-arclength continuation with a Newton corrector iteration. An iterative method, such as preconditioned bi-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.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B32 Bifurcations in context of PDEs
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