Numerical experiments with MG continuation algorithms

Abstract

This paper presents a systematic investigation of the numerical continuation algorithms for bifurcation problems (simple turning points and Hopf bifurcation points) of 2D nonlinear elliptic equations. The continuation algorithms employed are based only on iterative methods (Preconditioned Generalized Conjugate Gradient, PGCG, and Multigrid, MG). PGCG is mainly used as coarse grid solver in the MG cycle. Numerical experiments were made with the MG continuation algorithms developed by Hackbusch [W. Hackbusch, Multi-Grid Solution of Continuation Problems, Lecture Notes in Math., vol. 953, Springer, Berlin, 1982], Meis et al. [T.F. Meiss, H. Lehman, H. Michael, Application of the Multigrid Method to a Nonlinear Indefinite Problem, Lecture Notes in Math., vol. 960, Springer, Berlin, 1982], and Mittelmann and Weber [H.D. Mittelmann, H. Weber, Multi-grid solution of bifurcation problems, SIAM J. Sci. Statist. Comput. 6 (1985) 49]. The mathematical models selected, as test problems, are well-known diffusion–reaction systems; non-isothermal catalyst pellet and Lengyel–Epstein model of the CIMA reaction. The numerical methods proved to be efficient and reliable so that computations with fine grids can easily be performed.