Convergence to weak solutions of a space-time hybridized discontinuous Galerkin method for the incompressible Navier--Stokes equations

TL;DR

This paper proves that a space-time hybridized discontinuous Galerkin method for the Navier--Stokes equations converges to a weak solution, satisfying energy inequalities, as mesh size and time step go to zero.

Contribution

It introduces a convergence proof for a novel space-time hybridized discontinuous Galerkin method applied to the Navier--Stokes equations, utilizing discrete functional analysis tools.

Findings

Convergence to weak solutions as mesh size and time step decrease

Weak solutions satisfy the energy inequality

Application of discrete Aubin--Lions--Simon theorem

Abstract

We prove that a space-time hybridized discontinuous Galerkin method for the evolutionary Navier--Stokes equations converges to a weak solution as the time step and mesh size tend to zero. Moreover, we show that this weak solution satisfies the energy inequality. To perform our analysis, we make use of discrete functional analysis tools and a discrete version of the Aubin--Lions--Simon theorem.