PUB - Publikationen an der Universität Bielefeld

Let Ω⊂ℝ𝑛 be a convex polytope (𝑛≤3). The Ritz projection is the best approximation, in the 𝑊1,2 0-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 Ω 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.