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

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.

Figures (6)

• Figure 1: The elements of $\mathfrak{T}_T$ and $\mathcal{F}_{T}$.
• Figure 2: A closer look to the right part: The simplicial faces $\sigma_1,\sigma_3,\sigma_5$ belong to the set of interior faces $\mathfrak{F}_{T}^{{\rm i}}$ and we have $\sigma_2=F_1,\sigma_4=F_2,\sigma_6=F_3$, and $\sigma_7=F_4$.
• Figure 4: Cartesian.
• Figure 5: Hexagonal.
• Figure 6: Voronoi.

Theorems & Definitions (29)

• Remark 1: Comparison with the ${\boldsymbol{H}}_{\text{div}}({\Omega})$-conforming reconstruction of Castanon-Quiroz.Di-Pietro:23
• Lemma 2: Properties of ${\boldsymbol{R}}_{T}^k$
• Lemma 3: Raviart--Thomas--Nédélec lifting of the projection on $\boldsymbol{\mathcal{G}}^{{\rm c},k-1}(\mathfrak{T}_T)$
• proof : Proof of Lemma \ref{['lemm:rtn']}
• Lemma 4: Properties of $\|{\cdot}\|_{\text{R},T}$
• Remark 5: Norm $\|{\cdot}\|_{\text{R},h}$
• proof : Proof of Lemma \ref{['lem:ns:th']}
• Lemma 6: Properties of ${\varrho_{\mathfrak{T}_T}^{l+1}}$
• Lemma 7: Properties of ${{\mathfrak R}^{l}_{\boldsymbol{\mathcal{G}}^{},\mathfrak{T}_T}}$, ${\boldsymbol{\Gamma}_{\boldsymbol{\mathcal{G}}^{},\mathfrak{T}_T}^{l}}$, and ${\boldsymbol{\Gamma}_{\boldsymbol{\mathcal{G}}^{},\mathfrak{T}_T}^{{\rm c},l}}$
• proof