Pressure-robust and conforming discretization of the Stokes equations on anisotropic meshes

TL;DR

This paper introduces a conforming, pressure-robust discretization method for the Stokes equations on anisotropic meshes, overcoming previous limitations related to mesh shape-regularity and non-conforming elements.

Contribution

It develops a conforming pressure-robust discretization based on the Bernardi--Raugel element that works on anisotropic meshes, expanding the applicability of pressure-robust methods.

Findings

Numerical examples confirm the theoretical results.

The method achieves pressure-robustness on anisotropic meshes.

It extends existing approaches to conforming discretizations on irregular meshes.

Abstract

Pressure-robust discretizations for incompressible flows have been in the focus of research for the past years. Many publications construct exactly divergence-free methods or use a reconstruction approach [13] for existing methods like the Crouzeix--Raviart element in order to achieve pressure-robustness. To the best of our knowledge, except for our recent publications [3,4], all those articles impose a condition on the shape-regularity of the mesh, and the two mentioned papers that allow for anisotropic elements use a non-conforming velocity approximation. Based on the classical Bernardi--Raugel element we provide a conforming pressure-robust discretization using the reconstruction approach on anisotropic meshes. Numerical examples support the theory.