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Stability and Convergence of Mixed Finite Elements for Linear Regularized 13-Moment Equations

Shuang Hu, Huiteng Li, Zhenning Cai

Abstract

We present a stable and convergent mixed finite element method (MFEM) for the linear regularized 13-moment (R13) equations in rarefied gas dynamics. Unlike existing methods that require stabilization via penalty terms, our scheme achieves inherent stability by enriching the finite element basis with bubble functions. We provide a rigorous theoretical analysis, establishing second-order convergence rates in the $L^2$ norm under mild regularity assumptions. Beyond theoretical properties, our scheme demonstrates practical advantages over standard MFEM schemes, yielding robust numerical results even in the presence of geometric singularities.

Stability and Convergence of Mixed Finite Elements for Linear Regularized 13-Moment Equations

Abstract

We present a stable and convergent mixed finite element method (MFEM) for the linear regularized 13-moment (R13) equations in rarefied gas dynamics. Unlike existing methods that require stabilization via penalty terms, our scheme achieves inherent stability by enriching the finite element basis with bubble functions. We provide a rigorous theoretical analysis, establishing second-order convergence rates in the norm under mild regularity assumptions. Beyond theoretical properties, our scheme demonstrates practical advantages over standard MFEM schemes, yielding robust numerical results even in the presence of geometric singularities.
Paper Structure (19 sections, 18 theorems, 78 equations, 6 figures)

This paper contains 19 sections, 18 theorems, 78 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Regularized 13-moment system
  5. Weak Form
  6. Penalty-free mixed finite elements scheme
  7. Finite element spaces and Galerkin formulation
  8. Bubble-enriched element space and interpolation operator
  9. Kernel space and orthogonal decomposition
  10. Stability and convergence
  11. Discrete inf-sup condition
  12. Coercivity on the discrete kernel
  13. The a priori error estimate
  14. Numerical experiments
  15. Homogeneous flow around a cylinder
  16. ...and 4 more sections

Key Result

Lemma 3.3

The interpolation operator ${\@fontswitch{}{\mathcal{}} I}_h:H_0^1(\Omega;\mathbb{T})\rightarrow\Sigma_{h,0}$ defined by eq:interp_operator is $H^1$-stable and preserves the homogeneous trace. It has the following properties:

Figures (6)

  • Figure 1: The absolute $L^2$ error for each unknown in the R13 system.
  • Figure 2: Temperature and velocity distributions for Couette-Fourier flow around a cylinder.
  • Figure 3: Temperature distributions for Fourier flow in a square cavity.
  • Figure 4: Shear stress and heat distributions for the thermally-induced edge flow.
  • Figure 5: Velocity magnitude distributions for the thermally-induced edge flow.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Remark 3.1: Necessity for $m\geq d$
  • Remark 3.2: Why 2D
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5: Adjoint Operator
  • Proof 1
  • Lemma 3.6: Characterization of the Kernel
  • Proof 2
  • Lemma 3.7: Invariance of the Kernel
  • Proof 3
  • ...and 33 more