Pressure-robust error estimate of optimal order for the Stokes equations on domains with edges

TL;DR

This paper develops pressure-robust error estimates of optimal order for the Stokes equations on domains with edges, using a reconstruction approach to improve finite element methods and address singularities near edges.

Contribution

It introduces a reconstruction-based pressure-robust finite element method with optimal error estimates on anisotropic meshes for domains with edges.

Findings

Pressure-robust modified Crouzeix--Raviart method achieves optimal error rates.

Anisotropic mesh grading improves convergence near edges.

Numerical results confirm theoretical error estimates.

Abstract

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property for the discrete solution, by mapping discretely divergence-free test functions to exactly divergence-free functions in the sense of $\boldsymbol{H}(\operatorname{div})$. Moreover, the Stokes solution has locally singular behavior in three-dimensional domains near concave edges, which degrades the convergence rates on quasi-uniform meshes and makes anisotropic mesh grading reasonable in order to regain optimal convergence characteristics. Finite element error estimates of optimal order on meshes of tensor-product type with appropriate anisotropic grading are shown for the pressure-robust modified Crouzeix--Raviart method using the reconstruction approach. Numerical examples support the theoretical results.