A simple nonconforming tetrahedral element for the Stokes equations

TL;DR

This paper introduces a stable, low-order nonconforming tetrahedral element for the 3D Stokes equations that achieves similar accuracy to Taylor-Hood elements but with fewer degrees of freedom, and is suitable for stress formulations.

Contribution

It presents a novel nonconforming rotated bilinear tetrahedral element that is stable for the Stokes problem and compatible with stress-based formulations, reducing computational complexity.

Findings

The element is stable with a piecewise linear pressure approximation.

It satisfies Korn's inequality, ensuring stability in stress formulations.

It offers a computationally efficient alternative to traditional elements.

Abstract

In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in $\mathbb{R}^3$. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pressure. This gives an approximation similar to the well known continuous $P^2-P^1$ Taylor$-$Hood element, but with fewer degrees of freedom. The element is a stable non-conforming low order element which fulfils Korn's inequality, leading to stability also in the case where the Stokes equations are written on stress form for use in the case of free surface flow.