Error analysis of projection methods for non inf-sup stable mixed finite elements. The Navier-Stokes equations

TL;DR

This paper analyzes a modified projection method for the Navier-Stokes equations, demonstrating inherent stabilization that enables the use of non inf-sup stable finite elements without additional stabilization techniques.

Contribution

It introduces a modified Euler non-incremental scheme with inherent stabilization, extending the analysis to classical methods and relating them to pressure stabilized Petrov-Galerkin methods.

Findings

Error bounds for the modified scheme are established.

The scheme inherently stabilizes non inf-sup stable elements.

Classical methods are shown to have similar stabilization properties.

Abstract

We obtain error bounds for a modified Chorin-Teman (Euler non-incremental) method for non inf-sup stable mixed finite elements applied to the evolutionary Navier-Stokes equations. The analysis of the classical Euler non-incremental method is obtained as a particular case. We prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin-Temam method. The relation of the methods with the so called pressure stabilized Petrov Galerkin method (PSPG) is established. We do not assume non-local compatibility conditions for the solution.