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

    Torsten Linss, Natalia Kopteva, Goran Radojev, Martin Ossadnik

ETNA - Electronic Transactions on Numerical Analysis, pp. 99-122, 2024/02/29

doi: 10.1553/etna_vol60s99

doi: 10.1553/etna_vol60s99


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doi:10.1553/etna_vol60s99



doi:10.1553/etna_vol60s99

Abstract

A posteriori error estimates in the maximum norm are studied forvarious time-semidiscretisations applied to a class of linear parabolic equations. We summarise results from the literature and present some new improved error bounds. Crucial ingredients are certain bounds in the $L_1$-norm for the Green's function associated with the parabolic operator and its derivatives.

Keywords: parabolic problems, maximum-norm a posteriori error estimates, backward Euler, Crank–Nicolson, extrapolation, discontinuous Galerkin–Radau, backward differentiation formulae, Green's function