# Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

## TL;DR

This paper establishes new stability and localized error estimates for finite element solutions to the Stokes problem on convex polyhedra, extending known results to unstructured meshes in 2D and 3D.

## Contribution

It introduces novel stability and localization results for finite element approximations of the Stokes system on unstructured meshes, using regularized Green's functions.

## Key findings

- New stability estimates in $W^{1,inity}$ and $L^{inity}$ norms.
- Localized error estimates for finite element solutions on convex polyhedra.
- Extension of stability results to unstructured meshes in 2D and 3D.

## Abstract

The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in $W^{1,\infty}$ and $L^{\infty}$ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions.