Outline

The thesis will be laid out in the following way. In Chapter 2 we describe the elements that we build domains with. In Chapter 3 we will outline the polynomial basis functions that we use to approximate functions defined on these elements. There we will also show how these elements can be connected together so that their coordinate sysems line up. Chapter 4 details how we build operators on the elemental level and on the global level (i.e. over connected elements). In Chapters 5 and 6 we show how these building blocks can be applied to solving the incompressible Navier-Stokes equations and the equations of viscous magnetohydrodynamics. In Chapters 7 and 8 we show how the Discontinuous Galerkin Method, and its local operators can be used to solve the compressible Navier-Stokes equations and the compressible viscous Magnetohydrodynamic equations.

We have begun the unification of the Galerkin and Discontinuous Galerkin methods under the umbrella code . An outline of the current capability of this code is demonstrated in figure 1.3. All of the tree-root applications use the same set of code library routines and the same data structures.

  

Figure 1.3: Hierarchy of the code

In the applications chapters of this thesis we will discuss a number of benchmark tests and simulations using . For each simulation we will provide a table of parameters in order to reduce the amount of repetition involved. These tests are very important in determining that the code is correct and demonstrating that the underlying methods are robust and accurate when implemented.