On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

TL;DR

This paper explores the stability of bubble functions and introduces a new stabilized mixed finite element formulation for the Stokes problem, enabling equal order interpolation and highlighting limitations of existing methods.

Contribution

It presents a novel stabilized mixed formulation with a stability parameter derived via weighted residuals, and analyzes the relationship between stabilized and enriched finite element methods.

Findings

New stabilized formulation allows equal order interpolation for velocity and pressure.

Counterexample shows no direct equivalence between stabilized and enriched methods for certain elements.

Stability parameter derived purely from weighted residuals enhances formulation robustness.

Abstract

In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid-based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions.