A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

TL;DR

This paper introduces a low-cost, parameter-free enriched Galerkin method for the Stokes equations that is pressure-robust and computationally efficient, with proven optimal convergence and validated by numerical experiments.

Contribution

The paper presents a novel, parameter-free enriched Galerkin method with pressure-robustness and reduced computational effort for solving Stokes equations.

Findings

Achieves pressure-robustness via velocity reconstruction.

Reduces computational complexity by using weak derivatives.

Demonstrates optimal convergence rates in numerical tests.

Abstract

In this paper, we propose a low-cost, parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, the symmetric IPDG formulation, despite of its advantage of symmetry, requires a lot of computational effort to choose an optimal penalty parameter and to compute different trace terms. In order to reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a parameter-free numerical scheme which has reduced computational complexity as well as optimal rates of convergence. Moreover, we achieve pressure-robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.