Numerical experiments with MG continuation algorithms. (English) Zbl 1097.65064
Summary: 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 [W. Hackbusch [Lect. Notes Math. 953, 20–45 (1982; Zbl 0531.65052), T. F. Meis, H. Lehman and H. Michael [Lect. Notes Math. 960, 545–557 (1982; Zbl 0505.65050) and H. D. Mittelmann and H. Weber [SIAM J. Sci. Statist. Comput. 6, 49–60 (1985; Zbl 0557.65032)]. 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.
MSC:
| 65H10 | Numerical computation of solutions to systems of equations |
| 65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
Keywords:
multigrid; diffusion-reaction equation; numerical examples; finite difference method; continuation algorithms; bifurcation; turning points; Hopf bifurcation; nonlinear elliptic equations; preconditioned generalized conjugate gradientReferences:
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