In this talk I will first give a general introduction to the discontinuous
Galerkin finite element method and the main technical issues in generalizing
this method to solve PDEs with higher spatial derivatives. I will then
introduce the recent research of designing stable and convergent local
discontinuous Galerkin methods for solving various nonlinear dispersive
wave equations, including the Kadomtsev-Petviashvili equations and the
Zakharov-Kuznetsov equations. Numerical results will be shown to demonstrate
the good qualities of such methods. This is a joint work with Jue Yan and
Yan Xu.