An overlapping local projection stabilization for Galerkin approximations of Darcy and Stokes problems

TL;DR

This paper introduces a new local projection stabilization method for finite element approximations of Darcy and Stokes problems, ensuring stability and providing error estimates validated by numerical tests.

Contribution

The paper presents a novel stabilized finite element scheme using P1 elements for Darcy and Stokes problems, with proven stability and error bounds.

Findings

The stabilized scheme satisfies the inf-sup condition.

A priori error estimates are derived for both problems.

Numerical examples confirm the effectiveness of the stabilization.

Abstract

A priori analysis for a generalized local projection stabilized conforming finite element approximation of Darcy flow and Stokes problems is presented in this paper. A first-order conforming P1 finite element space is used to approximate both the velocity and the pressure. It is shown that the stabilized discrete bilinear form satisfy the inf-sup condition with respect to a generalized local projection norm. Moreover, a priori error estimates are derived for both problems. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.