Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods. (English) Zbl 1246.65178
The authors use a two-grid algorithm to linearize nonlinear reaction-diffusion equations discretized by a mixed finite element method. The key ingredient of the two-grid method is the use of one Newton iteration on the fine grid. It is shown that if the coarse grid, \(H\), and the fine grid, \(h\), satisfy \(H = O(h^{1/2})\) the two-grid algorithm can achieve the same accuracy as the mixed finite element solution. Generally speaking, different aspects of a complex problem can be treated by spaces of different scales. For the problem on hand, a very coarse grid space is sufficient for a nonlinear problem that is dominated by its linear part. The two-grid method provides a new approach to take advantage of some nice properties hidden in a complex problem. Results are confirmed numerically.
Reviewer: Rémi Vaillancourt (Ottawa)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
Keywords:
two-grid method; reaction-diffusion equations; mixed finite element methods; numerical examples; Newton iteration; algorithmReferences:
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