# A hybridizable discontinuous Galerkin method for the Navier--Stokes equations with pointwise divergence-free velocity field

## TL;DR

This paper presents a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations that ensures the velocity field is divergence-free at every point, improving stability and conservation properties.

## Contribution

It introduces a modified function space approach enabling a simple, divergence-free velocity approximation that is momentum conserving, energy stable, and pressure-robust.

## Key findings

- Method achieves pointwise divergence-free velocity fields.
- Numerical examples confirm stability and accuracy in 2D and 3D.
- Applicable for various polynomial approximation orders.

## Abstract

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.