Pointwise Gradient Estimate of the Ritz Projection

. Let \ (Rⁿ\) be a convex polytope (\ (n 3\) ). The Ritz projection is the best approximation, in the \ (W^1, 2₀\) -norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, the gradient at any point in \ (\) is controlled by the Hardy–Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces, and Lorentz spaces. Keywordsmaximal functiongradient estimatesMuckenhoupt weightsRitz projectionMSC codes65N3065N8065N12