Mimetic Finite Difference Method for Partial Differential Equations
TITLE: Mimetic Finite Difference Method for Partial Differential Equations
Speaker: Professor Hu Xiaozhe (Tufts University, U.S.A.)
ABSTRACT: Numerical simulation of partial differential equation (PDE) models is essential to understand complex applications in science and engineering. When developing discretization methods for solving these PDEs, it is essential to preserve the physical integrity of the problem to obtain high-fidelity approximations to the solutions. In this talk, we focus on the mimetic finite-difference (MFD) method, where discrete operators “mimic” their continuous counterparts, and physical properties and conservation laws are satisfied exactly on the discrete level. Using Maxwell’s equations as an example, we draw connections between the MFD method and finite-element exterior calculus, which provides a straightforward path to proving well-posedness, deriving error estimates, and developing robust linear solvers for MFD discretizations. We will also discuss MFD discretization for convection-dominated diffusion equations, which have a variety of applications in particle transport, electromagnetics, and magnetohydrodynamics. If time permits, we will briefly mention the generalizations of MFD to fractional PDEs and data-driven scientific machine learning.