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Research   [Back to home]

I work in the areas of computational and applied mathematics, and biophysical modeling and simulations, both of which involve analysis and numerical solutions of partial differential equations. Much of my work focus on the development of  high order numerical methods and the applications of these methods for producing high-fidelity computational simulations of molecular or cellular electrostatic potential, diffusion and reaction, signaling and dynamics.

Matched Interface and Boundary (MIB) Method

MIB is a high order algorithm for the numerical solutions of elliptic partial differential equations with discontinuous diffusion coefficients or source functions on irregular internal interfaces. These equations arise from various problems involving material interfaces: the Poisson-Boltzmann equation for electrostatic potential  in discontinuous dielectrics; the Maxwell equations or Helmholtz equation for the propagation of electromagnetic wave among different conductors; the Hele-Shaw problem modeling the crystal growth and solidification; and many others. Based on a standard central difference formulation on a regular Cartesian grid, MIB achieves its high accuracy and high order convergence by modifying the discretization schemes near the internal interface to implicitly enforce the continuity conditions.

Hybrid Finite Element and  Boundary Element Method

In the computational simulation of electrodiffusion with the Poisson-Nernst-Planck(PNP) equations, the Poisson equation is to be solved repeatedly for an steady-state or time-dependent electrostatic potential. My main research goal here is to devise and analyze an efficient and robust finite element/boundary element method for the solution of the Poisson equation with the discontinuous coefficient and singular sources, by integrating the Poincare-Steklov mapping, a fast Multipole method for the surface potential and the local surface/volumetric mesh refinement based on a posterior error estimates.

Local Spectral Method

Molecular Surface and Mesh Generation

Molecular Surface Generation with Diffusion Equation

Molecular Mesh Generation for Finite Element/Boundary Element Modeling

Biophysical Modeling and Simulations

Biomolecular Electrostatic Potential

Molecular/Cellular Signaling

Macromolecular Conformational Change