Nonconforming finite element method applied to the driven cavity problem

TL;DR

This paper introduces a minimal-parameter, stable nonconforming finite element method for accurately solving incompressible flow in a square cavity, effectively handling corner singularities without smoothing.

Contribution

It develops a new stable, low-degree-of-freedom finite element pair for the Stokes problem, optimized for computational efficiency and accuracy in cavity flow simulations.

Findings

Achieves high accuracy in cavity flow simulations.

Uses fewer degrees of freedom than existing methods.

Numerical results confirm the method's effectiveness.

Abstract

A cheapest stable nonconforming finite element method is presented for solving the incompressible flow in a square cavity without smoothing the corner singularities. The stable cheapest nonconforming finite element pair based on $P_1\times \widetilde{{\mathcal P}_{c}^h}$ on rectangular meshes \cite{stab-cheapest} is employed with a minimal modification for the discontinuous Dirichlet data on the top boundary, where $\widetilde{{\mathcal P}_{c}^h}$ is the finite element space of piecewise constant pressures with the globally one-dimensional checker-board pattern subspace eliminated. The proposed Stokes elements have least number of degrees of freedom compared to those of known stable Stokes elements. Three accuracy indications for our elements are analyzed and numerically verified. Also, various numerous computational results obtained by using our proposed element show excellent accuracy.