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A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids

Xuejun Xu, Shangyou Zhang

TL;DR

The paper introduces a stable 3D mixed finite element for the Stokes equations on general tetrahedral grids by constructing a nonconforming $P_3$ velocity space enriched with three inner $P_3^{nc}$ bubbles and six inner $P_4^{nc}$ bubbles per tetrahedron, where the bubbles have $P_2$-divergence and vanish $P_2$ moments on faces. When paired with a discontinuous $P_2$ pressure space, this $P_3^{nc}$-$P_2^{dis}$ element is inf-sup stable and yields a locally divergence-free discrete velocity, achieving quasi-optimal convergence. The authors prove stability and convergence results, provide a rigorous inf-sup analysis, and confirm the theory with numerical experiments on tetrahedral meshes. This work also argues that the proposed six $P_4^{nc}$ bubbles are the minimal and lowest-degree enrichment needed for stability on tetrahedra. Overall, the approach offers a low-order, robust, divergence-free scheme suitable for accurate Stokes simulations on general 3D meshes.

Abstract

A nonconforming $P_3$ finite element is constructed by enriching the conforming $P_3$ finite element space with three $P_3$ nonconforming bubbles and six additional $P_4$ nonconforming bubbles, on each tetrahedron. Here the divergence of the $P_4$ bubble is not a $P_3$ polynomial, but a $P_2$ polynomial. This nonconforming $P_3$ finite element, combined with the discontinuous $P_2$ finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special $P_4$ bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.

A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids

TL;DR

The paper introduces a stable 3D mixed finite element for the Stokes equations on general tetrahedral grids by constructing a nonconforming velocity space enriched with three inner bubbles and six inner bubbles per tetrahedron, where the bubbles have -divergence and vanish moments on faces. When paired with a discontinuous pressure space, this - element is inf-sup stable and yields a locally divergence-free discrete velocity, achieving quasi-optimal convergence. The authors prove stability and convergence results, provide a rigorous inf-sup analysis, and confirm the theory with numerical experiments on tetrahedral meshes. This work also argues that the proposed six bubbles are the minimal and lowest-degree enrichment needed for stability on tetrahedra. Overall, the approach offers a low-order, robust, divergence-free scheme suitable for accurate Stokes simulations on general 3D meshes.

Abstract

A nonconforming finite element is constructed by enriching the conforming finite element space with three nonconforming bubbles and six additional nonconforming bubbles, on each tetrahedron. Here the divergence of the bubble is not a polynomial, but a polynomial. This nonconforming finite element, combined with the discontinuous finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.
Paper Structure (5 sections, 5 theorems, 42 equations, 2 figures, 1 table)

This paper contains 5 sections, 5 theorems, 42 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The ${\mathbf{B}}_4$ bubbles and the mixed $P_3^{{\operatorname{\tiny nc}}}$-$P_2^{{\operatorname{\tiny dis}}}$ finite element
  3. The stability and convergence
  4. Numerical computation
  5. Ethical Statement

Key Result

Lemma 2.1

The bilinear form $(\nabla {\mathbf{u}}_h, \nabla {\mathbf{v}}_h)$ is positive definite on the basis of ${\mathbf{V}}_h$, defined in V-h.

Figures (2)

  • Figure 1: The unit reference tetrahedron $\hat{T}$, an affine mapping ${\mathbf{F}}_1$ and a general tetrahedron $T_{1672}$ (on a unit cube).
  • Figure 2: The first two tetrahedral grids for the computation in Tables \ref{['t1']}.

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof