Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier--Stokes Equations with Local Projection Stabilization

TL;DR

This paper analyzes the error behavior of non inf-sup stable finite element methods with local projection stabilization for the time-dependent Navier--Stokes equations, providing error estimates and numerical validation.

Contribution

It offers new error estimates for LPS-stabilized finite element methods applied to Navier--Stokes, including decay rates and effects of stabilization terms, independent of viscosity.

Findings

Velocity error decays at rate l+1/2 when viscosity is small.

Error bounds are independent of inverse powers of viscosity.

Numerical results confirm theoretical error estimates.

Abstract

This paper studies non inf-sup stable finite element approximations to the evolutionary Navier--Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants in the error bounds are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty(0,T;L^2(\Omega))$ decays with rate $l+1/2$ in the case that $\nu\le h$, with $\nu$ being the dimensionless viscosity and $h$ the mesh width. In the analysis of another method, it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results.