Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations

TL;DR

This paper studies a class of divergence-free HDG methods for stationary Navier-Stokes equations, providing theoretical analysis and numerical validation of their accuracy and stability.

Contribution

It introduces a divergence-free HDG method with specific finite element spaces and proves optimal error estimates and stability conditions.

Findings

The methods are pressure-robust and divergence-free.

Optimal error estimates are established.

Numerical experiments confirm theoretical results.

Abstract

This paper analyzes a class of globally divergence-free (and therefore pressure-robust) hybridizable discontinuous Galerkin (HDG) finite element methods for stationary Navier-Stokes equations. The methods use the $\mathcal{P}_{k}/\mathcal{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for the velocity and pressure approximations in the interior of elements, and piecewise $\mathcal{P}_k/\mathcal{P}_{k}$ for the trace approximations of the velocity and pressure on the inter-element boundaries. It is shown that the uniqueness condition for the discrete solution is guaranteed by that for the continuous solution together with a sufficiently small mesh size. Based on the derived discrete HDG Sobolev embedding properties, optimal error estimates are obtained. Numerical experiments are performed to verify the theoretical analysis.