| Over the last decade, Discontinuous Galerkin methods have been found very attractive for solving numerically non-linear hyperbolic equations. They can easily handle complex geometries since they are finite element methods. They provide high-order accurate approximations, in contrast to traditional finite volume methods. Because of their local nature, they have a high degree of parallelism and are suitable for adaptivity. In recent years, there has been a tremendous interest in extending these type of methods to problems where the diffusion is not negligible. In this thesis, we study the convergence properties of the hp-version of the so called Local Discontinuous Galerkin (LDG) finite element method for time-dependent convection-diffusion problems. We also provide the first a priori error analysis of this method for a model elliptic problem. These theoretical results are tested in a series of numerical experiments on one and two dimensional domains. Finally, we provide the first numerical comparison of the LDG method with the other discontinuous Galerkin methods developed for elliptic problems. We also describe an object oriented code for two dimensional domains for a general class of discontinuous Galerkin methods. This code can use arbitrary polynomial degree, can handle unstructured conforming triangular meshes and is able to perform local refinement and unrefinement. We conclude with some future directions of research. |
