Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

TL;DR

This paper develops a new a posteriori error estimator for pressure-robust Stokes finite element methods, focusing on the divergence-free part of the source term to improve robustness and efficiency, especially in pressure-dominant cases.

Contribution

It introduces a novel error estimator based on the curl of the right-hand side, enhancing pressure-robustness and reliability of error control in Stokes FEM.

Findings

The estimator is reliable and efficient for pressure-robust methods.

Numerical tests confirm improved error estimation in pressure-dominant scenarios.

The approach is applicable to Taylor--Hood and mini finite element methods.

Abstract

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods.