Note on Multigrid Methods for Nonlinear Problems

摘要

Well-known methods for solving discretized nonlinear partial differentialequations using multigrid techniques are: Nonlinear Multi-Grid Method (NMGM); The Full Approximation Scheme (FAS); and Newton iteration combined with a linear multigrid method. In section 1, we describe the Nonlinear Two-Grid Method (NTGM), the two grid FAS, and the relation between these two. In section 2, we introduce a model class of locally affine problems. In section 3, we recall the principle of Nested Iteration, which is useful for generating acceptable starting vectors. In section 4, it is shown that for our model class of nonlinear problems the NTGM can be derived in a natural way, such that the role of the parameters ocurring in the method is clear. In section 5, the Newton iteration combined with a linear multigrid method is given. Finally in section 6, methods are compared.