Discontinuous Galerkin Finite Element Methods (DGFEM) were introduced over 25 years ago for the numerical solution of first-order hyperbolic problems. Although, subsequently much of the research of partial differential equations has concentrated on the development and the analysis of conforming Finite Element Methods (FEM), recent years have witnessed renewed interest in discontinuous schemes. In contrast to standard FEM, DGFEM allow discontinuous numerical solutions. The DGFEM can also be thought of as the higher-order extension of the classical cell centre Finite Volume Method (FVM)—a popular discretization technique in the computational aerodynamics community.

This Diploma thesis is devoted to the implementation of the hp-version of the DGFEM as presented in [1]. The motivation is to have a numerical evidence for the proof of exponential convergence in a polygon [2]. The implemented second-order partial differential equation covers a large class of equations which includes advection-dominated diffusion problems and problems of elliptic type. The proof of the exponential convergence does not include the advection term, tough.

Unfortunately, there is an important difference between the implemented DGFEM from [1] and the one used in [2]: the former uses weakly enforced Dirichlet boundary conditions and the latter strongly enforced Dirichlet boundary conditions.

The outline of the thesis is as follows: chapter 1 gives a very short introduction to FEM and then to DGFEM including the variational formulation of the implemented equation. In chapter 2, a summary of the results in [2] is given (the proofs are omitted). Chapter 3 presents the basis of the software which was used (Concepts by Dr. Christian Lage). In chapter 4, the needed extensions to Concepts are explained. And in chapter 5, the numerical results are presented. In the appendix, some of the source code is given for the experienced reader.

PDF of my diploma thesis.

1. E. Süli, P. Houston and C. Schwab (1999), hp-Finite Element Methods for Hyperbolic Problems, Tech. Rep. 99-14, Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zürich.
2. T. P. Wihler, P. Frauenfelder and C. Schwab (2003). Exponential Convergence of the hp-DGFEM for Diffusion Problems, Comput. Math. Appl., Vol. 46, Nr. 1 (2003), 183-205.
3. Concepts Homepage.