An Adaptive Weak Galerkin Method for Multi-Scale Reaction--Convection--Diffusion Systems in Chemical Applications

This paper presents a novel computational framework for the numerical solution of multi-parameter singularly perturbed reaction--convection--diffusion problems that arise frequently in chemical modeling applications. We develop an \((hp)\)-adaptive Weak Galerkin finite element method that operates on anisotropic meshes, specifically designed to handle the intricate boundary layers, interior layers, and evolving patterns that characterize chemical systems such as electrochemical cells and excitable media. The method incorporates three key innovations: a stable weak formulation tailored for multi-parameter problems, a robust a posteriori error estimator in a chemically-informed balanced norm that properly weights errors in critical regions, and an adaptive algorithm that simultaneously performs anisotropic \((h)\)-refinement and \((p)\)-enrichment based on local solution properties. Numerical experiments demonstrate the method's effectiveness in resolving electrochemical boundary layers without non-physical oscillations, tracking rotating chemical waves in excitable media, and outperforming state-of-the-art approaches in both accuracy and computational efficiency. The proposed method achieves exponential convergence rates for problems with complex layer structures while maintaining robust performance across a wide range of parameter values. This work provides chemists and computational researchers with a powerful tool for simulating multi-scale phenomena in electrochemical systems, pattern formation, and reaction--diffusion processes that were previously computationally prohibitive.