Research

Quantum Scientific Computing and Numerical Methods for PDEs 

Quantum Numerical Methods for PDEs

• Quantum Multigrid and Preconditioning

• Quantum-Native Numerical Methods for PDEs (QnFEM on GitHub)

• Quantum Linear Solvers and Complexity

Structure-Preserving Numerical PDEs

• Transport and Convection-Dominated Problems

• Incompressible and Porous-Media Flow (EG-Fluid on GitHub)

• High-Dimensional and Spatiotemporal Discretizations

Selected Applications and Previous Projects

• PDE-Constrained Optimization

• Machine-Learning-Assisted Singularity Detection

• Nonlinear Beam Models

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

• J. H. Adler, Z. Ding, X. Hu, S. Lee, and W. Pan. Quantum multigrid preconditioning via oracle-free block-encoding of expanded systems. To be submitted (2026).

• S. Lee and L. Mu. Fully discrete analysis of pressure-robust enriched Galerkin methods for the time-dependent Stokes equations. Submitted to Computers & Mathematics with Applications (2026).

• J. H. Adler, X. Hu, and S. Lee. A unified spatiotemporal differential-form framework for time-dependent convection-diffusion problems. Part I: Continuous formulation and analysis. Submitted to Computers & Mathematics with Applications (2026).

• S. Jeong, S. Lee, and S. Liu. A monotone finite element scheme preserving desired-state bounds in convection-dominated optimal control problems. Journal of Computational and Applied Mathematics, 490 (2026) 117959.

• 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. Journal of Scientific Computing, 107, 52 (2026).

• 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 & 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.

• Quantum implementation of geometric multigrid methods via expanded systems. (with J. H. Adler, Z. Ding, X. Hu, and W. Pan)

• Quantum linear solver complexity for adaptive finite-element discretizations (with J. H. Adler and X. Hu)

• An incidence-based edge-averaged method for convection-diffusion problems on polytopal meshes (with J. H. Adler, X. Hu, and E. O'Keefe)