A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation

TL;DR

This paper introduces a pressure-robust finite element discretization for Oseen's equations, utilizing a novel stabilization based on the vorticity equation that improves accuracy and stability at high Reynolds numbers.

Contribution

The authors develop a pressure-robust stabilization method for Oseen's equations that is independent of pressure gradients, providing optimal error estimates and improved numerical performance.

Findings

Achieved an $O(h^{k+1/2})$ error estimate in the $L^2$-norm.

Numerical results confirm theoretical error bounds.

Method outperforms classical SUPG stabilization in tests.

Abstract

Discretization of Navier-Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressure-independent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an $O(h^{k+\frac12})$ error estimate in the $L^2$-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.