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Stability of the Ritz projection in weighted $W^{1,1}$

Irene Drelichman, Ricardo G. Duran

TL;DR

This work establishes stability of the Ritz projection in weighted $W^{1,1}$ spaces for finite element approximations of the Poisson problem on convex polyhedral domains with quasi-uniform meshes and weights $w\in A_1$. It extends prior $W^{1,p}_w$ stability results to the endpoint case $p=1$ by adapting the previous proof and employing a decomposition of the derivative of the Ritz projection into a distributional and a Green's-function term, controlled through the Hardy--Littlewood maximal operator and uniform Green's function bounds. The main outcome is the a priori estimate $\| abla R_h u\|_{L^1_w(\Omega)} \lesssim \|\nabla u\|_{L^1_w(\Omega)}$ for all $u\in W^{1,1}_0(\Omega)$ and $w\in A_1$, enabling robust stability for finite element Poisson solvers in singular settings. This strengthens the theoretical foundation for FEM in weighted, potentially singular regimes and informs numerical analyses where $L^1$-type control is critical.

Abstract

We prove the stability in weighted $W^{1,1}$ spaces for standard finite element approximations of the Poisson equation in convex polygonal or polyhedral domains, when the weight belongs to Muckenhoupt's class $A_1$ and the family of meshes is quasi-uniform.

Stability of the Ritz projection in weighted $W^{1,1}$

TL;DR

This work establishes stability of the Ritz projection in weighted spaces for finite element approximations of the Poisson problem on convex polyhedral domains with quasi-uniform meshes and weights . It extends prior stability results to the endpoint case by adapting the previous proof and employing a decomposition of the derivative of the Ritz projection into a distributional and a Green's-function term, controlled through the Hardy--Littlewood maximal operator and uniform Green's function bounds. The main outcome is the a priori estimate for all and , enabling robust stability for finite element Poisson solvers in singular settings. This strengthens the theoretical foundation for FEM in weighted, potentially singular regimes and informs numerical analyses where -type control is critical.

Abstract

We prove the stability in weighted spaces for standard finite element approximations of the Poisson equation in convex polygonal or polyhedral domains, when the weight belongs to Muckenhoupt's class and the family of meshes is quasi-uniform.
Paper Structure (3 sections, 1 theorem, 13 equations)

This paper contains 3 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Stability in weighted $W^{1,1}_0(\Omega)$

Key Result

Theorem 3.1

Let $\Omega \subset \mathbb{R}^2$ or $\mathbb{R}^3$ be a convex polytope and $\mathbb{T}=\{\mathcal{T}_h\}_{h>0}$ be a family of conforming and quasi-uniform triangulations of $\Omega$. For every $u \in W^{1,1}_0(\Omega)$ and every weight $w\in A_1$, there holds

Theorems & Definitions (2)

  • Theorem 3.1
  • proof