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 $L^2$ 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.