A posteriori error analysis for discontinuous Galerkin methods for time discrete semilinear parabolic problems

A posteriori error estimates for time discretisations for semilinear parabolic (evolution) problems by the discontinuous Galerkin method DG(r) of arbitrary order r >= 0 are derived and analysed. The semilinear evolutionary problems of the form u' + Au = f(u) with A is either linear or monotone gamma(2)-angle bounded operator are considered. The main tool in this analysis is the time reconstruction function (U)over cap> of the approximate discrete solution U of the exact solution u. Two classes of nonlinearities are addressed: firstly, when the source term is globally Lipschitz continuous and secondly when the source term is locally Lipschitz continuous.