$P_1$-nonconforming divergence-free finite element method on square meshes for Stokes equations

TL;DR

This paper introduces a locally divergence-free subspace within the $P_1$-nonconforming finite element method on square meshes, enabling efficient and decoupled solutions for velocity and pressure in Stokes equations.

Contribution

It presents a novel divergence-free subspace that simplifies the linear system and decouples velocity and pressure computations for Stokes equations.

Findings

Smaller linear systems compared to traditional methods

Rapid explicit computation of pressure after velocity

Stable solution for Stokes equations on square meshes

Abstract

Recently, the $P_1$-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its locally divergence-free subspace to solve the elliptic problem for the velocity only decoupled from the Stokes equation. The concerning system of linear equations is much smaller compared to the Stokes equations. Furthermore, it is split into two smaller ones. After solving the velocity first, the pressure in the Stokes problem can be obtained by an explicit method very rapidly.