A Reynolds-semi-robust method with hybrid velocity and pressure for the unsteady incompressible Navier--Stokes equations

TL;DR

This paper introduces a hybrid finite element method for unsteady incompressible Navier-Stokes equations that is both pressure-robust and Reynolds-quasi-robust, with validated convergence and numerical experiments.

Contribution

It presents a novel hybrid discretization approach that maintains robustness against pressure variations and low viscosity effects in Navier-Stokes simulations.

Findings

Method is pressure-robust, unaffected by irrotational forces.

Error estimates do not depend on inverse viscosity for smooth solutions.

Numerical experiments confirm theoretical convergence rates.

Abstract

In this paper we propose and analyze a new Finite Element method for the solution of the two- and three-dimensional incompressible Navier--Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds-quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting pre-asymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments.