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A nonconforming P2 and discontinuous P1 mixed finite element on tetrahedral grids

Shangyou Zhang

TL;DR

This work introduces a stable, nonconforming $P_2$ velocity space on tetrahedral grids by enriching the conforming $P_2$ space with seven $P_2^{\mathrm{nc}}$ bubble functions, paired with a discontinuous $P_1$ pressure space to solve the stationary Stokes equations. The $P_2^{\mathrm{nc}}$/$P_1^{\mathrm{dis}}$ element space is carefully constructed so that the velocity space $V_h$ is $\,\big(\sum c_i \phi_i\big) \oplus (P_2^{\mathrm{c}} \cap H_0^1(\Omega)^3)$ and the pressure space is $P_h = P_1^{\mathrm{dis}}$, with a proven discrete inf-sup condition and linear independence of the enrichment. A constructive stability proof, leveraging Scott–Zhang interpolation and a bubble-based divergence correction, yields mesh-size independent inf-sup stability. Convergence analysis shows quasi-optimal rates: $\|\mathbf{u}-\mathbf{u}_h\|_{1,h} + \|p-p_h\|_0 \le C h^2 (|\mathbf{u}|_3 + |p|_2)$ and $\|\mathbf{u}-\mathbf{u}_h\|_0 \le C h^3 (|\mathbf{u}|_3 + |p|_2)$, with numerical tests on the unit cube validating these rates and confirming optimal performance. This work resolves a long-standing question about stability for the 3D $P_2^{\mathrm{nc}}$/$P_1^{\mathrm{dis}}$ Stokes element on general tetrahedral meshes and provides a practically implementable, provably stable method.

Abstract

A nonconforming $P_2$ finite element is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently such a mixed finite element method produces optimal-order convergen solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

A nonconforming P2 and discontinuous P1 mixed finite element on tetrahedral grids

TL;DR

This work introduces a stable, nonconforming velocity space on tetrahedral grids by enriching the conforming space with seven bubble functions, paired with a discontinuous pressure space to solve the stationary Stokes equations. The / element space is carefully constructed so that the velocity space is and the pressure space is , with a proven discrete inf-sup condition and linear independence of the enrichment. A constructive stability proof, leveraging Scott–Zhang interpolation and a bubble-based divergence correction, yields mesh-size independent inf-sup stability. Convergence analysis shows quasi-optimal rates: and , with numerical tests on the unit cube validating these rates and confirming optimal performance. This work resolves a long-standing question about stability for the 3D / Stokes element on general tetrahedral meshes and provides a practically implementable, provably stable method.

Abstract

A nonconforming finite element is constructed by enriching the conforming finite element space with seven nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming finite element, combined with the discontinuous finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently such a mixed finite element method produces optimal-order convergen solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.
Paper Structure (14 sections, 5 theorems, 71 equations, 2 figures, 1 table)

This paper contains 14 sections, 5 theorems, 71 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The $P_2^{{\operatorname{nc}}}$-bubbles and the $P_2^{{\operatorname{nc}}}$/$P_{1}^{{\operatorname{dis}}}$ element
  3. The stability
  4. The convergence
  5. Numerical tests
  6. Ethical Statement
  7. Compliance with Ethical Standards
  8. Funding
  9. Conflict of Interest
  10. Ethical approval
  11. Informed consent
  12. Availability of supporting data
  13. Authors' contributions
  14. Acknowledgments

Key Result

Lemma 2.1

The $P_2^{{\operatorname{nc}}}$ bubble space and the $P_2^{{\operatorname{c}}}$ vector space in V-h are linearly independent. In other words, the following $n_{nc}+n_c$ vectors are linearly independent, where $\phi_i$ and $\psi_j$ are defined in g-2 and g-1, respectively.

Figures (2)

  • Figure 1: A tetrahedron $T$, its vertices ${\mathbf{x}}_i$ and its barycentric coordinates $\lambda_i$.
  • Figure 2: The first three 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 4.1
  • proof
  • Theorem 4.2
  • proof