SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1212-1225, June 2024.
Abstract. Let [math] be a convex polytope ([math]). The Ritz projection is the best approximation, in the [math]-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 [math] 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.

中文翻译：

---

Ritz 投影的逐点梯度估计


《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1212-1225 页，2024 年 6 月。

抽象的。设 [math] 为凸多面体 ([math])。在数学范数中，里兹投影是有限元空间中给定函数的最佳近似。当这种有限元空间基于准均匀三角剖分构建时，我们会显示 Ritz 投影的逐点估计。也就是说，[math] 中任意点的梯度由同一点原始函数梯度的 Hardy-Littlewood 极大函数控制。根据这一估计，紧随偏微分方程分析中感兴趣的各种空间上的里兹投影的稳定性。其中包括加权空间、Orlicz 空间和洛伦兹空间。