The system of regularized 13-moment (R13) equations can be viewed as a generalization of the classical Navier-Stokes-Fourier equations that includes non-conservative moments such as the stress tensor and heat flux, so that non-equilibrium effects such as Knudsen boundary layers and anti-Fourier flows can be described. While the framework of finite element methods has been extensively studied for Navier-Stokes equations, the counterpart for R13 equations remains to be studied. The state-of-the-art finite element method to solve linear R13 equations is to apply the Continuous Interior Penalty method to suppress the spurious oscillation in the numerical solution, which requires selection of stabilization parameters that may be problem-dependent [1]. In this work, we develop a parameter-free finite element method based on the recent progress on the well-posedness of the steady-state linear R13 equations [2]. New finite elements are constructed to fulfil the inf-sup condition and coercivity required by the convergence theory, and our experiments show that oscillations no longer appear in our numerical results. REFERENCES [1] L. Theisen and M. Torrilhon. 2021. FenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform. ACM Trans. Math. Softw. 47, 2, Article 17 (June 2021), 29 pages. https://doi.org/10.1145/3442378. [2] P. Lewintan, L. Theisen, and M. Torrilhon. Well-Posedness of the R13 Equations Using Tensor-Valued Korn Inequalities. ArXiv: 2501.14108.