Research

What makes a numerical simulation faithful to the physical world?

In numerical analysis and scientific computing, traditional methods have emphasized stability and accuracy, often assuming that sufficiently refined meshes ensure reliability. Yet real phenomena are high-dimensional and computationally demanding, and in extreme regimes, fine meshes alone cannot guarantee meaningful results.

My research addresses this challenge by developing structure-preserving methods that respect the underlying physical laws, together with efficient discretizations for large-scale systems. These principles guide my work on transport, fluid dynamics, and multiphysics PDEs.

Research Projects

Current / Main Projects

Previous / Side Projects

Grants

NSF DMS-2513394, Computational Mathematics Program, "Finite Elements in the Quantum Era", August 2025 - July 2028. Co-PI (with J. Adler and X. Hu)

Publications

J. Adler, X. Hu, and S. Lee. A unified spatiotemporal formulation with physics-preserving structures for time-dependent convection-diffusion problems. Submitted (2025).
S. Jeong, S. Lee, and S. Liu. A monotone finite element scheme preserving desired-state bounds in convection-dominated optimal control problems. Submitted (2025).
S. Jeong, S. Lee, and K. Wang. A C0 weak Galerkin method with preconditioning for optimal control problems with general tracking and pointwise state constraints. Under revision for JSC (2025).
S. Cao, L. Chen, and S. Lee. Edge-averaged virtual element methods for convection-diffusion and convection-dominated problems. Journal of Scientific Computing, 104, 70 (2025).
Z. Chen, S. Lee, and L. Mu. Automated detection and characterization of singularities in functions using neural networks-from FFT signals. International Journal of Numerical Analysis and Modeling, 21(5) (2024) 629-651.
S. Lee and L. Mu. A uniform and pressure-robust enriched Galerkin method for the Brinkman equations. Journal of Scientific Computing, 99, 39 (2024).
S. Lee and L. Mu. A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations. Computers and Mathematics with Applications, 166 (2024) 51-64.
X. Hu, S. Lee, L. Mu, and S.-Y. Yi. Pressure-robust enriched Galerkin methods for the Stokes equations. Journal of Computational and Applied Mathematics, 436 (2024) 115449.
J. Ahn, S. Lee, and E.-J. Park. C0 interior penalty methods for a dynamic nonlinear beam model. Applied Mathematics and Computation, 339 (2018) 685-700.

In Preparation

Structure-preserving numerical methods for spatiotemporal convection-diffusion problems. (with J. Adler and X. Hu)
Randomized solvers for nonsymmetric M-matrices with quantum-inspired speedup. (with J. Adler and X. Hu)
Stable lowest-order finite element schemes for the time-dependent Stokes equations. (with L. Mu)

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