New stabilized $P_1\times P_0$ finite element methods for nearly inviscid and incompressible flows

TL;DR

This paper introduces a new stabilized finite element method for incompressible flows that remains accurate and pressure-robust even at very low viscosities, without needing parameter tuning.

Contribution

It presents a novel stabilized $P_1\times P_0$ finite element scheme based on a reduced Bernardi--Raugel element with static face bubble condensation, offering uniform error estimates and improved robustness.

Findings

Error estimates are uniform with respect to viscosity.

The method is pressure-robust in the small viscosity regime.

Validated on benchmark problems in 2D and 3D.

Abstract

This work proposes a new stabilized $P_1\times P_0$ finite element method for solving the incompressible Navier--Stokes equations. The numerical scheme is based on a reduced Bernardi--Raugel element with statically condensed face bubbles and is pressure-robust in the small viscosity regime. For the Stokes problem, an error estimate uniform with respect to the kinematic viscosity is shown. For the Navier--Stokes equation, the nonlinear convection term is discretized using an edge-averaged finite element method. In comparison with classical schemes, the proposed method does not require tuning of parameters and is validated for competitiveness on several benchmark problems in 2 and 3 dimensional space.