# Stability of the Stokes projection on weighted spaces and applications

## TL;DR

This paper proves the stability of the finite element Stokes projection on weighted spaces with Muckenhoupt weights in convex polytopes, enabling improved error estimates for Stokes problems with singular sources.

## Contribution

It establishes stability results for the Stokes projection on weighted spaces with variable integrability, extending previous work to more general weights and dimensions.

## Key findings

- Stability of the Stokes projection on weighted spaces in 2D and 3D.
- Application of stability results to error estimation for singular source problems.
- Extension of stability analysis to weights in the Muckenhoupt class.

## Abstract

We show that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces $\mathbf{W}^{1,p}_0(\omega,\Omega) \times L^p(\omega,\Omega)$, where the weight belongs to a certain Muckenhoupt class and the integrability index can be different from two. We show how this estimate can be applied to obtain error estimates for approximations of the solution to the Stokes problem with singular sources.