Two low-order nonconforming finite element methods for the Stokes flow in 3D

TL;DR

This paper introduces two novel low-order nonconforming finite element methods for 3D Stokes flow, extending previous 2D approaches and establishing their mathematical properties for stability and well-posedness.

Contribution

The paper develops two new 3D nonconforming FEMs for Stokes flow, proving their stability via discrete Korn and inf-sup conditions, and highlights limitations of direct generalizations from 2D methods.

Findings

Proved uniform discrete Korn inequality and inf-sup condition.

Established well-posedness of the proposed FEMs.

Demonstrated non-existence of direct 3D generalizations of 2D FEMs.

Abstract

In this paper, we propose two low order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the non-conforming FEM of Kouhia and Stenberg (1995, Comput. Methods Appl. Mech. Engrg.). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf-sup condition hold uniformly in the meshsize and also for a non-empty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia-Stenberg FEM to three space dimensions: The finite element space with one non-conforming and two conforming piecewise affine components does not satisfy a discrete inf-sup condition with piecewise constant pressure approximations, while finite element functions with two non-conforming and one conforming component do not satisfy a discrete Korn inequality.