A Reynolds-semi-robust and pressure robust Hybrid High-Order method for the time dependent incompressible Navier--Stokes equations on general meshes

TL;DR

This paper introduces a novel Hybrid High-Order discretization for the incompressible Navier-Stokes equations that is robust against Reynolds number variations and pressure, achieving optimal convergence rates on general meshes.

Contribution

The paper presents a Reynolds-semi-robust, pressure-robust HHO method with a new penalty term and subgrid projection, improving accuracy in convection-dominated flows.

Findings

Achieves $h^{k+1/2}$ convergence order in velocity error norms.

Demonstrates robustness with respect to Reynolds number.

Validates effectiveness on polygonal meshes in 2D simulations.

Abstract

In this work we develop and analyze a Reynolds-semi-robust and pressure-robust Hybrid High-Order (HHO) discretization of the incompressible Navier--Stokes equations. Reynolds-semi-robustness refers to the fact that, under suitable regularity assumptions, the right-hand side of the velocity error estimate does not depend on the inverse of the viscosity. This property is obtained here through a penalty term which involves a subtle projection of the convective term on a subgrid space constructed element by element. The estimated convergence order for the $L^\infty(L^2)$- and $L^2(\text{energy})$-norm of the velocity is $h^{k+\frac12}$, which matches the best results for continuous and discontinuous Galerkin methods and corresponds to the one expected for HHO methods in convection-dominated regimes. Two-dimensional numerical results on a variety of polygonal meshes complete the exposition.