On mesh independence and Newton-type methods

Abstract

Mesh-independent convergence of Newton-type methods for the solution of nonlinear partial differential equations is discussed. First, under certain local smoothness assumptions, it is shown that by properly relating the mesh parameters $H$ and $h$ for a coarse and a fine discretization mesh, it suffices to compute the solution of the nonlinear equation on the coarse mesh and subsequently correct it once using the linearized (Newton) equation on the fine mesh. In this way the iteration error will be of the same order as the discretization error. The proper relation is found to be $H=h^1^/^\alpha$where in the ideal case, $\alpha=4$. This means that in practice the coarse mesh is very coarse. To solve the coarse mesh problem it is shown that under a Hölder continuity assumption, a truncated and approximate generalized conjugate gradient method with search directions updated from an (inexact) Newton direction vector, converges globally, i.e. independent of the initial vector. Further, it is shown that the number of steps before the superlinear rate of convergence sets in is bounded, independent of the mesh parametr.