A unified approach to maximum-norm a posteriori error estimation for second-order time discretisations of parabolic equations

Abstract

A class of linear parabolic equations are considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretisations combined with finite element discretisations in space. In particular we study the Crank-Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 2 (BDF-2), the Lobatto IIIC method and a two-stage SDIRK method. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.