A simple finite element method for the Stokes equations

TL;DR

This paper introduces a straightforward finite element method for solving Stokes and Navier-Stokes equations using piecewise constant approximations for velocity and pressure, simplifying implementation and analysis.

Contribution

It presents a novel simple primal velocity-pressure finite element method with divergence-free basis construction, reducing the saddle point problem to a symmetric positive definite system.

Findings

Method is accurate and robust in numerical experiments.

Reduces computational complexity by simplifying the saddle point problem.

Provides error analysis and implementation insights.

Abstract

The goal of this paper is to introduce a simple finite element method to solve the Stokes and the Navier-Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate and robust.