Analysis of an exactly mass conserving space-time hybridized discontinuous Galerkin method for the time-dependent Navier--Stokes equations

TL;DR

This paper introduces and analyzes a space-time hybridized discontinuous Galerkin method for the Navier--Stokes equations, emphasizing mass conservation, stability, and pressure robustness, with proven existence, uniqueness, and error estimates.

Contribution

It presents a novel numerical scheme with proven properties for the Navier--Stokes equations, including existence, uniqueness in 2D, and error bounds.

Findings

Solution exists for the nonlinear system in 2D and 3D.

Unique solution in 2D under small data.

A priori error estimates for velocity.

Abstract

We introduce and analyze a space-time hybridized discontinuous Galerkin method for the evolutionary Navier--Stokes equations. Key features of the numerical scheme include point-wise mass conservation, energy stability, and pressure robustness. We prove that there exists a solution to the resulting nonlinear algebraic system in two and three spatial dimensions, and that this solution is unique in two spatial dimensions under a small data assumption. A priori error estimates are derived for the velocity in a mesh-dependent energy norm.