A quadrilateral 'mini' finite element for the Stokes problem using a single bubble function

TL;DR

This paper introduces a simplified quadrilateral 'mini' finite element for the Stokes problem that uses only one bubble function per element, ensuring stability and improving upon previous methods that required multiple bubble functions.

Contribution

The paper demonstrates that a single bubble function suffices for the quadrilateral 'mini' finite element, simplifying the implementation and enhancing stability for solving Stokes equations.

Findings

Single bubble function ensures inf-sup stability

Improves previous quadrilateral 'mini' element methods

Simplifies finite element implementation for Stokes problem

Abstract

We consider a quadrilateral 'mini' finite element for approximating the solution of Stokes equations using a quadrilateral mesh. We use the standard bilinear finite element space enriched with element-wise defined bubble functions for the velocity and the standard bilinear finite element space for the pressure space. With a simple modification of the standard bubble function we show that a single bubble function is sufficient to ensure the inf-sup condition. We have thus improved an earlier result on the quadrilateral 'mini' element, where more than one bubble function are used to get the stability.