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

TL;DR

This paper introduces a stable mixed finite element method for the linear regularized 13-moment equations in rarefied gas dynamics, achieving convergence without penalty stabilization by enriching the basis with bubble functions.

Contribution

The paper presents a novel MFEM scheme that is inherently stable and convergent for R13 equations, avoiding penalty stabilization and handling geometric singularities effectively.

Findings

Achieves second-order convergence in L2 norm.

Demonstrates robustness in complex geometries.

Provides rigorous theoretical stability analysis.

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.