3. Differential Equations

In this chapter discretization schemes for differential equations are discussed in the context of the formalism introduced in the previous chapter. By now it was shown how functions are introduced to the computer, how functions can be evaluated, added and transformed to other bases. The main focus of the following sections is to show that all discretization schemes which are used for the discretization of differential equations can be formalized in the same manner using the formalusm presented in the previous chapter.

As will be discussed in Chapter 4, the outcome of a discretization scheme is an equation system, where the number of equations equals the dimension of the used function space. In order to fulfill this requirement an association scheme for equations and unknown variables is introduced. This association is based on the consideration that each basis function has one equation on which the coefficient of the respective function has the most influence. Such considerations are also referred to as control functions.

In the following sections the methods of formula specification of Chapter 2 are used in order to form expressions according to the respective discretization scheme. It shall be shown that typical discretization schemes can be specified using the formalism introduced. Moreover, the formalism implies a view on how the discretization scheme directly influences the underlying data structure. Each discretization scheme imposes different requirements on the underlying data structures with respect to traversal and storage of the quantities. A discussion on how the data structures can be chosen in an appropriate manner can be seen in Section 2.5.

In each of the sections the Laplace equation on a two-dimensional triangularly tessellated simulation domain is investigated and formulated.

Subsections

• 3. Differential Equations
  • 3. 1 Finite Element Schemes
    • 3. 1. 1 Weak Formulation and Galerkin Schemes
    • 3. 1. 2 Re-Formulation on Topological Properties
    • 3. 1. 3 Integration Example
    • 3. 1. 4 Conclusion
  • 3. 2 Finite Volume Schemes
    • 3. 2. 1 Secondary Graph Method - Gauß Integral Theorem
    • 3. 2. 2 Edge-Based Boundary Integrals
    • 3. 2. 3 Topological Structure of the Finite Volume Scheme
    • 3. 2. 4 Laplace Equation
    • 3. 2. 5 Volume Integration
    • 3. 2. 6 Drift-Diffusion Semiconductor Equations
    • 3. 2. 7 Conclusion
  • 3. 3 Boundary Element Method
    • 3. 3. 1 Standard Formulation
    • 3. 3. 2 Implementation-Based Formulation
    • 3. 3. 3 Conclusion
  • 3. 4 Finite Difference Schemes
    • 3. 4. 1 Conventional Method
    • 3. 4. 2 Topological Neighborhood Considerations
    • 3. 4. 3 Laplace Operator
    • 3. 4. 4 Conclusion
  • 3. 5 Final Considerations