Pointwise gradient estimate of the ritz projection

TL;DR

This paper establishes a pointwise gradient estimate for the Ritz projection in convex polytopes, linking the gradient to the Hardy--Littlewood maximal function, which ensures stability across various function spaces.

Contribution

It provides a novel pointwise gradient estimate for the Ritz projection on convex polytopes, extending stability results to weighted, Orlicz, and Lorentz spaces.

Findings

Gradient at any point controlled by Hardy--Littlewood maximal function

Stability of Ritz projection in diverse function spaces

Applicable to finite element spaces on quasiuniform triangulations

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.