1. 1. 2 Modeling

The solution of each engineering problem requires knowledge in a field such as applied physics, chemistry, process technology, civil engineering, construction, medicine, or any other. Modeling is required in order to describe the physical effects relevant for answering the engineering problem. All effects which have to be taken into consideration are comprised by the model chosen. A typical result of the modeling process is the mathematical formulation of the desired physical phenomena, for instance as partial differential equation.

In general, many different models are available for the solution of a problem which differ in fidelity and complexity, each of them offering a tradeoff between complexity and fidelity of the solution obtained by the model. For this reason different levels of detail are employed depending on the phenomena that are interesting to the preceding engineering problem.

The implications of mathematical considerations on the model have to be kept to a minimum and should only affect aspects such as overall solvability and perhaps uniqueness of the problem. At this stage it is not necessary to define the discretization scheme as well as the required algebraic solution methods. A premature restriction to a discretization and an algebraic method unnecessarily leads to limitations of usability and applicability, and other mathematical methods which are developed later on cannot be incorporated into the original program.

Example: It can be shown that (non-relativistic) electrostatic problems (engineering) in general result in diagonally dominant symmetric matrix problems. In fact two different problems have been solved in one step, namely the modeling of the physical behavior of a given configuration using electrostatic equations and the subsequent discretization of the resulting differential equations (elliptic self-adjoint equations). Between these steps an interface is used, namely the continuous problem formulation. At this interface a separation between modeling and the further processing of the model equation(s) can be established. The advantage of this separation is that the details of the model and the further processing steps can be considered as black-boxes.