Dr. Gie’s scholarly work lies at the intersection of singular perturbation theory, fluid mechanics, and numerical analysis. His research addresses boundary, corner, and interior layer phenomena arising in viscous and diffusive limits, and develops analytical and computational tools that remain robust in singular regimes.

In recent years, his work has expanded to include structure-preserving and analysis-informed learning methods for stiff partial differential equations, integrating asymptotic analysis with conservative numerical discretizations and machine-learning-based components.