A Unified Framework of Stabilized Finite Element Methods for Solving the Boltzmann Transport Equation

Abstract

This technical note presents a unified framework of stabilized finite element methods for solving the Boltzmann transport equation. The unified framework is derived from the standard Galerkin weak form with a subgrid scale model, which is different from the traditional Petrov-Galerkin finite element framework that modifies the test function to construct the stabilization term. By this method, first, the unknowns are decomposed into their numerical solutions and residuals. The decomposed unknowns are then embedded into the Galerkin weak form with an approximation for the residual, which yields a stabilized variational formula. Different methods of stabilization are derived from different approximations of the residual. Under this framework, all the frequently used stabilized methods can be obtained, including the streamline upwinding Petrov-Galerkin method, the Galerkin least-squares method, and the algebraic subgrid scale method. Thus, a unified framework of such methods is established. The similarities and differences across the different approximations are also compared in this technical note. The numerical results show that the behaviors of different methods are similar with the same stabilization parameters and that all these stabilized techniques can yield satisfactory and stable solutions.

Keywords:

• Boltzmann transport equation
• stabilized finite element methods
• approximate residual equation
• stabilization parameter

Acknowledgments

The authors want to thank Yifei Li in Xi’an Jiaotong University for formatting-related writing assistance.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is financially supported by the National Nature Science Foundation of China [grant number 11735011]; and National Key R&D Program of China [grant number 2017YFE0302200].