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Maximal regularity is a kind of a priori estimates for parabolic-typeequations and it plays an important role in the theory of nonlineardifferential equations. The aim of this paper is to investigate the temporallydiscrete counterpart of maximal regularity for the discontinuous Galerkin (DG)time-stepping method. We will establish such an estimate without logarithmicfactor over a quasi-uniform temporal mesh. To show the main result, weintroduce the temporally regularized Green's function and then reduce thediscrete maximal regularity to a weighted error estimate for its DGapproximation. Our results would be useful for investigation of DGapproximation of nonlinear parabolic problems.

Discrete maximal regularity for the discontinuous Galerkin time-stepping method without logarithmic factor