Pressure-robust enriched Galerkin methods for the Stokes equations

TL;DR

This paper introduces a pressure-robust enriched Galerkin method for the Stokes equations, enhancing accuracy and efficiency by employing velocity reconstruction, static condensation, and viscosity-independent preconditioners, validated through numerical experiments.

Contribution

It develops a pressure-robust EG scheme with a velocity reconstruction operator and analyzes a stabilized version with static condensation, improving robustness and computational efficiency.

Findings

Velocity error is independent of pressure and viscosity.

The stabilized method allows static condensation of velocity components.

Preconditioners perform well regardless of viscosity.

Abstract

In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]. The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized $H^1$-conforming $\mathbb{P}_1$-$\mathbb{P}_0$ method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.