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

Technical Summary: A Pressure-Robust HHO Method for the Incompressible Navier–Stokes Equations on General Meshes

Problem Statement
The paper addresses the numerical approximation of the incompressible Navier–Stokes equations, specifically focusing on the challenge of achieving pressure robustness. In standard discretizations, the error estimates for the velocity field often depend on the pressure, leading to poor performance when the body force contains large irrotational components (i.e., gradients of a scalar potential). While pressure-robust methods have been developed for standard finite elements and matching simplicial meshes, extending these properties to general polytopal meshes (supporting arbitrary polyhedral elements and non-matching interfaces) remains a significant challenge. The authors aim to resolve an open question from their previous work [11]: whether a pressure-robust Hybrid High-Order (HHO) method can be constructed for general polytopal meshes.

Methodology
The proposed method is a fully pressure-robust HHO scheme valid in two and three dimensions. The core innovation lies in the construction of a local divergence-preserving velocity reconstruction ($R^k_T$) defined on a subtriangulation of each mesh element $T$.

1. Mesh and Submesh: The method operates on a polyhedral mesh $\mathcal{M}_h$. Crucially, it assumes the existence of a matching simplicial submesh $\mathfrak{T}_T$ within each element $T$. A specific geometric assumption is made that all simplices in $\mathfrak{T}_T$ share a common vertex ($x_T$), allowing for pyramidal or non-pyramidal submeshes.
2. Divergence-Preserving Reconstruction: For a given discrete velocity unknown $\mathbf{v}_T$, the reconstruction $R^k_T \mathbf{v}_T$ is obtained by solving a local mixed problem on the submesh $\mathfrak{T}_T$. This problem seeks a function in the Raviart–Thomas–Nédélec space $RT_k(\mathfrak{T}_T)$ that:
   • Matches the normal components of the discrete velocity on the element boundaries.
   • Preserves the discrete divergence (commuting with the discrete divergence operator $D^k_T$).
   • Satisfies orthogonality conditions with respect to the Koszul space $G_{c, k-1}(T)$ and the interior of the element.
   • This construction avoids the limitations of previous approaches (e.g., reliance on Wachspress coordinates) that required restrictive face geometries or led to open theoretical problems regarding gradient bounds.
3. Discrete Formulation:
   • Viscous Term: Uses the standard HHO gradient reconstruction on the submesh.
   • Body Force: The forcing term $\mathbf{f}$ is discretized by testing against the reconstructed velocity $R^k_h \mathbf{v}_h$ rather than the raw discrete unknown. This leverages the divergence-preserving property to ensure that irrotational forces (gradients) do not affect the velocity solution.
   • Convective Term: A novel trilinear form is introduced using the reconstructed velocity $R^k_h$ and an enriched gradient reconstruction $G^{2(k+1)}_{\mathfrak{T}_T}$. This formulation ensures non-dissipativity and consistency while maintaining pressure robustness.

Key Contributions

• General Mesh Support: The paper provides a positive answer to the extension of pressure-robust HHO methods to general polytopal meshes, overcoming previous restrictions to simplicial or specific polyhedral geometries.
• Novel Reconstruction: The introduction of a divergence-preserving velocity reconstruction based on a local mixed problem on a subtriangulation. This approach is shown to be well-posed for arbitrary polynomial degrees $k \ge 0$ and general polytopes, provided the submesh satisfies the common-vertex assumption.
• Theoretical Analysis: The authors establish rigorous error estimates. They prove that the velocity error is independent of the pressure and the irrotational part of the body force (parameter $\lambda$ in the Hodge decomposition). The convergence rate is $O(h^{k+1})$ in the energy norm and $O(h^{k+2})$ in the $L^2$-norm for the velocity, under standard regularity assumptions.
• Robustness Proof: Theoretical results demonstrate that the method is robust with respect to large irrotational body forces, a property not shared by standard HHO schemes.

Numerical Results
The theoretical findings are validated through a comprehensive set of numerical tests in two dimensions:

1. Kovasznay Flow: Convergence rates are verified on Cartesian, hexagonal, and Kershaw meshes. The results confirm optimal convergence orders ($h^{k+1}$ for energy norm, $h^{k+2}$ for $L^2$ velocity error) for both the proposed method and standard HHO.
2. Robustness Test: A problem with a large irrotational body force ($\lambda = 10^6$) is solved. The proposed method reproduces the exact solution (for $k=1$) or converges rapidly regardless of the magnitude of $\lambda$. In contrast, the standard HHO method of [8] exhibits large errors that do not diminish with mesh refinement, confirming its lack of pressure robustness.
3. Lid-Driven Cavity: The method is applied to the classical lid-driven cavity problem at Reynolds number $Re=1000$. The results align with reference solutions from the literature. Furthermore, when an irrotational body force is added, the velocity profiles remain unchanged, whereas the standard method converges to a different, incorrect solution.

Significance and Claims
The paper claims to provide a fully pressure-robust HHO method that works on general meshes composed of polytopal elements. The significance lies in bridging the gap between the flexibility of polytopal methods (handling complex geometries and non-matching interfaces) and the physical fidelity of pressure-robust schemes (correctly handling irrotational forces). The authors emphasize that their construction relies on a local mixed problem on a subtriangulation, which allows them to derive necessary continuity and approximation bounds without the geometric restrictions (e.g., triangular faces only) or open theoretical issues (e.g., Wachspress coordinate bounds) that plagued earlier polyhedral approaches. The work extends the applicability of pressure-robust discretizations to a broader class of meshes, maintaining arbitrary-order accuracy.