On Counterexamples for Rates of Convergence concerning Numerical Solutions of Initial Value Problems

Abstract

The present note is concerned with the approximation of the exact solution of a properly posed initial value problem by finite difference methods. It is shown that those rates of convergence obtained in previous papers on abstract Lax-type theorems with rates are indeed sharp. This is achieved as a consequence of a general uniform boundedness principle with rates, given in the setting of discrete approximations of Banach spaces. The method of proof is the familiar gliding hump method but now equipped with rates. The applications presented emphasize the unifying approach to various concrete results scattered in the literature.