Higher-order discontinuous Galerkin time discretizations the evolutionary Navier--Stokes equations

TL;DR

This paper develops higher-order discontinuous Galerkin methods for time discretization of the Navier--Stokes equations, providing optimal error bounds and numerical validation for both velocity and pressure.

Contribution

It introduces higher-order discontinuous Galerkin temporal discretizations with error analysis and stabilization techniques for the Navier--Stokes equations.

Findings

Optimal error bounds for velocity independent of viscosity

Error estimates for velocity and pressure in fully discrete case

Numerical results confirm theoretical predictions

Abstract

Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier--Stokes equations. The spatial discretization based on inf-sup stable pairs of finite element spaces is stabilised using a one-level local projection stabilisation method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for the semi-discrete case. For the fully discrete case, error estimates for both velocity and pressure are given. Numerical results support the theoretical predictions.