Characteristic Methods and Locally Divergence-free Finite Elements

This talk is divided into two parts.
Convection-diffusion equations arise from many applications. Solutions to these problems usually exhibit sharp fronts and even shocks, which pose serious challenges to existing numerical methods. In the first part, we will present the characteristic methods that efficiently solve convection-diffusion problems. The characteristic methods can be combined with finite elements or wavelets to accurately resolve sharp fluid fronts. Numerical results of applying the characteristic methods to the kinematics of two-dimensional resistive magnetohydrodynamic flows will be presented. "Divergence-free" is an important physical property that should be respected by numerical methods. Locally divergence-free (LDF) finite elements has gained researchers' attention recently.
In the 2nd part, we examine the approximation properties of the LDF finite elements and their use in the nonconforming or discontinuous Galerkin formulations. Preliminary numerical results on applying the LDF finite elements to 3-dimensional Maxwell source problems and eigenvalue problems will be presented. We will also discuss applications of the LDF finite elements to the Navier-Stokes equation.