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

Abstract

Abstract A class of linear parabolic equations is considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretizations combined with finite element discretizations in space. In particular, we study the Crank–Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 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.