CDR-Solv: Solving the Convection-Diffusion-Reaction Equation with Algebraic Sub-grid Scale Stabilization using Python

This investigation developed and validated a comprehensive computational framework for solving convection-diffusion-reaction (CDR) equations under convection-dominated conditions through the implementation of Algebraic Sub-grid Scale (ASGS) stabilization coupled with high-order finite elements. The research addressed the fundamental challenge of spurious oscillations that emerge in standard Galerkin formulations when convection significantly exceeds diffusion, a prevalent issue in transport phenomena modeling. A novel Python-based educational software platform, designated CDR-Solv, was engineered to demonstrate the effectiveness of ASGS stabilization across polynomial degrees ranging from linear to cubic elements. Numerical experiments conducted with low diffusion coefficients demonstrated that the ASGS method successfully eliminated numerical instabilities while maintaining solution accuracy across diverse source term configurations. The stabilization parameter τK proved instrumental in achieving computational stability without compromising mathematical rigor. Comparative analysis revealed that higher-order elements consistently outperformed linear approximations in capturing boundary layer phenomena and sharp gradient regions. The primary contribution of this work lies in the development of an open-source educational platform that democratizes access to advanced stabilization techniques while providing transparency in algorithmic implementation. The CDR-Solv framework enables researchers and educators to systematically explore the relationship between polynomial degree selection and solution quality in convection-dominated regimes.