A mesh independence principle for inexact Newton-like methods and their discretizations under generalized Lipschitz conditions

摘要

In this manuscript, we study inexact Newton-like methods for the solution of nonlinear operator equations in a Banach space involving a nondifferentiable term, and their discretized versions in connection with the mesh independence principle. This principle asserts that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform (or not) discretization. In some of our earlier papers we extended these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use inexact Newton-like methods and even more general conditions. Our results, on the one hand, extend, and on the other hand, make more practical and applicable previous results. Moreover, we can solve a wider range of problems and find sharper error bounds. Several applications of our results to two point boundary value problems are illustrated.