A sharp discrete maximal regularity for the discontinuous Galerkin time-stepping method
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Maximal regularity is a kind of a priori estimates for parabolic-type equations and plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish an estimate sharper than the existing discrete maximal regularity, and our estimate is the best possible result. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. We will also obtain an optimal order error estimate for the DG time-stepping method as an application.
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