A unified framework of continuous and discontinuous Galerkin methods for solving the incompressible Navier--Stokes equation

TL;DR

This paper introduces a comprehensive numerical framework unifying various Galerkin methods for solving the incompressible Navier--Stokes equations, demonstrating stability and competitiveness through extensive numerical experiments.

Contribution

It develops a unified approach for multiple Galerkin methods with pressure robustness, stability proofs, and a comparative analysis with classical schemes.

Findings

The scheme is stable under implicit Runge--Kutta time discretization.

Numerical results show competitive performance on benchmark problems.

The framework unifies different Galerkin methods with a common penalty term discussion.

Abstract

In this paper, we propose a unified numerical framework for the time-dependent incompressible Navier--Stokes equation which yields the $H^1$-, $H(\text{div})$-conforming, and discontinuous Galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness. Under minimum assumption on Galerkin spaces, the semi- and fully-discrete stability is proved when a family of implicit Runge--Kutta methods are used for time discretization. Furthermore, we present a unified discussion on the penalty term. Numerical experiments are presented to compare our schemes with classical schemes in the literature in both unsteady and steady situations. It turns out that our scheme is competitive when applied to well-known benchmark problems such as Taylor--Green vortex, Kovasznay flow, potential flow, lid driven cavity flow, and the flow around a cylinder.