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In this paper we establish a best approximation property of fully discreteGalerkin finite element solutions of second order parabolic problems on convexpolygonal and polyhedral domains in the $L^\infty$ norm. The discretizationmethod uses of continuous Lagrange finite elements in space and discontinuousGalerkin methods in time of an arbitrary order. The method of proof differsfrom the established fully discrete error estimate techniques and for the firsttime allows to obtain such results in three space dimensions. It uses ellipticresults, discrete resolvent estimates in weighted norms, and the discretemaximal parabolic regularity for discontinuous Galerkin methods established bythe authors in [16]. In addition, the proof does not require any relationshipbetween spatial mesh sizes and time steps. We also establish a local bestapproximation property that shows a more local behavior of the error at a givenpoint.

Pointwise best approximation results for Galerkin finite element solutions of parabolic problems