A pressure-robust HHO method for the solution of the incompressible Navier-Stokes equations on general meshes

TL;DR

This work advances pressure robustness for incompressible Navier–Stokes discretizations on general polyhedral meshes by introducing a divergence-preserving velocity reconstruction computed locally on element submeshes. The reconstruction is integrated into the discretization of the body force and the convective term, yielding velocity error estimates that are independent of pressure and irrotational forcing. The authors develop a full Hybrid High-Order (HHO) framework with a divergence-compatible velocity reconstruction, a gradient reconstruction on a submesh, and robust discrete forms for viscous, convective, and body-force operators, proving well-posedness and convergence for polynomial degrees $k\ge 0$. Numerical tests in 2D on polygonal meshes confirm high-order convergence and, crucially, robustness to large irrotational body forces, matching reference solutions and outperforming non-pressure-robust schemes on challenging meshes.

Abstract

In a recent work [10], we have introduced a pressure-robust Hybrid High-Order method for the numerical solution of the incompressible Navier-Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for polynomial degrees $k \geq 0$ and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples.

Figures (9)

• Figure 1: The elements of $\mathfrak{T}_{T}$ and $\mathcal{F}_{T}$.
• Figure 2: A closer look to the bottom part: The faces $\sigma_1,\sigma_2,\sigma_3$ are interior faces, i.e., $\{\sigma_1,\sigma_2,\sigma_3\}\subset\mathfrak{F}_{T}^{{\rm i}}$. For the set $\{\sigma_4,\sigma_5\}$, we have $\sigma_4=F_1$ and $\sigma_5=F_2$.
• Figure 4: Pyramidal submesh.
• Figure 5: Non-pyramidal submesh.
• Figure 7: Cartesian.

Theorems & Definitions (24)

• Remark 1: Allowing more than pyramidal meshes
• Lemma 2: Properties of $\boldsymbol{R}_{T}^k$
• Lemma 3: Raviart--Thomas lifting of the projection in $\boldsymbol{\mathcal{G}}^{{\rm c},k-1}(T)$
• Remark 4: The common vertex assumption for $k\in\{0,1\}$
• proof
• Remark 5: Convexity assumption
• Proposition 6: Sobolev inequalities for the velocity reconstruction
• proof
• Lemma 7: Properties of $\boldsymbol{G}_{\mathfrak{T}_{T}}^{l}$
• proof