Introduction

Abstract

Discontinuous Galerkin (DG) methods are a class of finite-element methods using completely discontinuous basis functions, which are usually chosen as piece-wise polynomials. Since the basis functions can be completely discontinuous, these methods have the flexibility which is not shared by typical finite-element methods, such as the allowance of arbitrary triangulation with hanging nodes, complete freedom in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and extremely local data structure (elements only communicate with immediate neighbors regardless of the order of accuracy of the scheme) and the resulting embarrassingly high parallel efficiency (usually more than 99% for a fixed mesh, and more than 80% for a dynamic load balancing with adaptive meshes which change often during time evolution), see, e.g. [5]. A very good example to illustrate the capability of the discontinuous Galerkin method in h-p adaptivity, efficiency in parallel dynamic load balancing, and excellent resolution properties is the successful simulation of the Rayleigh-Taylor flow instabilities in [38].