Maximum-norm a posteriori error bounds for an extrapolated Euler/finite element discretisation of parabolic equations

Abstract

A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green’s function of the parabolic operator.AMS subject classification (2000): 65M15, 65M60.

A review of maximum-norm a posteriori error bounds for time-semidiscretisations of parabolic equations

“…We follow [15] and consider a piecewise linear reconstruction Û of the approximations U j , j = 0, 1, . .…”

“…Following [7] and [5], the authors of the present study have published a number of results on residual-type a posteriori error estimates in the maximum norm for parabolic equations utilising and merging various approaches and considering various classes of temporal discretisation [5,6,11,12,15,17]. In this survey, we review these results in a unified manner.…”