Nonlinear Elliptic Problems Falls Multigrid Method

The cascadic multigrid method has been shown to be one of the most efficient iterative techniques for solving large boundary value problems, the main advantage of which is coarse-grid-correction free, and as a result it can be viewed as a one-way multigrid method. In this paper, based on a two-grid approach a multilevel linearization approach and a new cascadic multigrid method is proposed for the second order nonlinear elliptic boundary value problems. We provide the error analysis and the computation complexity of the algorithms including the case of the fixed grid level and the case of the arbitrary grid level. When the number of grid level is fixed, a superconvergence result for the multilevel linearization algorithm is established. Onthe other hand, with traditional iterations and the conjugate gradient(CG) as smoothers, we can show the optimal convergence rate of the cascadic method in energy norm for 1-D and 2-D cases. When the mesh level is arbitrary, we use a duality argument and obtain the quasi-optimality of the algorithm only for 2-D problems. Thus when we solve nonlinear elliptic problems with this algorithm, the complexity for linear and nonlinear problems is essentially equal. Finally the numerical experiments also show the effectiveness of the method.