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

TL;DR

This paper analyzes modified projection methods for the transient Stokes problem, demonstrating inherent stabilization that enables the use of non inf-sup stable finite elements without extra stabilization, supported by numerical tests.

Contribution

It introduces modified projection schemes with inherent stabilization for non inf-sup stable elements, extending classical methods and relating to pressure stabilized Petrov-Galerkin techniques.

Findings

Modified Euler schemes have inherent stabilization.

Classical Chorin-Temam method also stabilizes non inf-sup elements.

Numerical tests confirm theoretical stabilization results.

Abstract

A modified Chorin-Teman (Euler non-incremental) projection method and a modified Euler incremental projection method for non inf-sup stable mixed finite elements are analyzed. The analysis of the classical Euler non-incremental and Euler incremental methods are obtained as a particular case. We first 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. For the second scheme, we study a stabilization that allows the use of equal-order pairs of finite elements. The relation of the methods with the so called pressure stabilized Petrov Galerkin method (PSPG) is established. The influence of the chosen initial approximations in the computed approximations to the pressure is analyzed. Numerical tests confirm the theoretical results.