Survey and Recent Development for Discontinuous Galerkin Methods

Chi-Wang Shu
Division of Applied Mathematics
Brown University

Discontinuous Galerkin finite element methods have been under rapid
development over the past few years for solving time dependent
convection dominated PDEs, with applications to computational fluid
dynamics and other physical and engineering areas.  In this talk I
will give a brief overview of discontinuous Galerkin methods for
solving hyperbolic partial differential equations (joint work with
Cockburn et al.), and present recent results on developing local
discontinuous Galerkin methods for solving time dependent partial
differential equations containing second spatial derivatives (convection
diffusion equations, joint work with Cockburn), and third and higher
spatial derivatives (KdV type equations, time dependent bi-harmonic type
equations, etc., joint work with Yan, Levy and Xu).  We also present recent
results of locally divergence-free discontinuous Galerkin methods for
solving Maxwell equations, ideal magnetohydrodynamics (MHD) equations
and Hamilton-Jacobi equations (joint work with Li and Cockburn).