2. Discrete Problems and Discretization

Methods designed to transform continuous mathematical problems - in most cases partial differential equations - into algebraic problems are called discretization methods. Such a continuous problem is usually given by a model for the governing physical processes which are determined by the physical reality to be investigated. The problem is given on a continuous space which is usually a subset of $\mathcal{R}, \mathcal{R}^2, \mathcal{R}^3, \ldots$ . This means that one is faced with at least an ordinary differential equation (ODE) or a partial differential equation (PDE) that optionally contains temporal derivatives.

Discretization considers the fact that analytical methods which are known from functional analysis cannot be directly applied to functions stored in a computer, and an assumption on the functional structure of the single elements has to be made.

One main aspect of this chapter is the development of a formalism which is designed for writing expressions in the context of discretization. It is shown that such a formalism can be used for a concise definition of the topological as well as the functional structure of the respective discretization formulae. This formalism can be used for the specification of computer programs, when such a formalism contains sufficient information to carry out all operations which are required for the evaluation of the formulae.

Subsections
• 2. Discrete Problems and Discretization
  • 2. 1 Numerical Discretization
    • 2. 1. 1 Numerical Discretization Error
    • 2. 1. 2 Errors of a Numerically Discrete System
  • 2. 2 Functional Discretization
    • 2. 2. 1 Functions and Discrete Representations
    • 2. 2. 2 Spatial Resolution of a Simulation Domain
    • 2. 2. 3 Shape Functions and Basis Functions
  • 2. 3 Topological Structures
    • 2. 3. 1 Finite Cell Complexes
    • 2. 3. 2 Incidence and Traversal
    • 2. 3. 3 Orientation
  • 2. 4 Topological Mappings of Shape Functions
    • 2. 4. 1 Geometrical Properties of the Elements
    • 2. 4. 2 Association of Functions and Topological Elements
  • 2. 5 Formulation of Discrete Problems
    • 2. 5. 1 Basis Operations
    • 2. 5. 2 Constants and Second Order Functions
    • 2. 5. 3 Accumulation
    • 2. 5. 4 Accumulation Methods
    • 2. 5. 5 Multi-Argument Functions
    • 2. 5. 6 Accumulation in Depth
  • 2. 6 Examples
    • 2. 6. 1 Topological Calculations
      • 2. 6. 2 Number of Neighboring Edges of a Vertex
    • 2. 6. 3 Discrete Solution Methods
    • 2. 6. 4 Geometric Examples
    • 2. 6. 5 Quadrature of Discretized Functions