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arXiv · 2305.03575

Pointwise gradient estimate of the ritz projection

Lars Diening·Julian Rolfes·Abner J. Salgado
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Abstract

Let $\Omega \subset \mathbb{R}^n$ be a convex polytope ($n \leq 3$). The Ritz projection is the best approximation, in the $W^{1,2}_0$-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, that the gradient at any point in $\Omega$ 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.

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Lars Diening, Julian Rolfes, Abner J. Salgado. 2023-05-05. Pointwise gradient estimate of the ritz projection. https://arxiv.org/abs/2305.03575

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