Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier-Stokes equations

TL;DR

This paper establishes pressure-robust and convection-robust error bounds for divergence-free finite element methods applied to the incompressible Navier-Stokes equations, ensuring accurate velocity approximation in convection-dominated flows.

Contribution

It proves an $O(h^{k+1/2})$ error estimate for velocity in the $L^2$ norm that is independent of pressure and Reynolds number, advancing finite element analysis.

Findings

Error bound of $O(h^{k+1/2})$ for velocity in convection-dominated regime

Velocity error bound is pressure robust

Constants in error bounds are independent of Reynolds number

Abstract

In this paper we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1/2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust (the constants in the error bounds are independent of the Reynolds number).