Some remarks on the instability of approximate solutions for ODEs

Abstract

We show a strange property of an approximate solution for the initial value problem of ODEs. This study is originally motivated by the numerical verification methods of solutions for parabolic initial-boundary value problems. We prove that an a priori norm estimate of the derivative of approximation for simple initial value problems could not be stable, even though the concerned numerical scheme is a natural finite element approximation. Since the corresponding estimates for the exact solution is bounded by a given function, it implies that the discretization could no longer maintain the same property at all. Therefore, this result should be an interesting example which shows an essential gap between the infinite and finite dimensional problems.