Pointwise gradient estimate of the ritz projection

DOI10.1137/23M1571800arXiv2305.03575MaRDI QIDQ6435510

Abner J. Salgado, Lars Diening, Julian Rolfes

Publication date: 5 May 2023

Abstract: Let $Omegasubsetmathbb{R}^n$ be a convex polytope ($nleq3$). The Ritz projection is the best approximation, in the $W_0^{1,2}$-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.

Mathematics Subject Classification ID

Maximal functions, Littlewood-Paley theory (42B25) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Fundamental solutions, Green's function methods, etc. for boundary value problems involving PDEs (65N80)

Related Items (1)

Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes

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