## 1. 1. 5 Discrete Problems

After the discretization of a differential equation has been performed, the equation is reduced to a system of algebraic equations. In general, there are three main classes of discrete problems which are commonly known in scientific computing:

• Linear equation systems
• Nonlinear equation systems
• Eigenvalue equation systems

Apart from finiteness, discrete problems which are the result of discretization in general have the same algebraic structure of their associated problems. This means that in most cases a linear problem (e.g. a linear differential equation) results in a linear equation system and a continuous eigenvalue problem becomes discrete but remains an eigenvalue problem.

The discrete problem formulation is an interface between the discretization scheme and the algebraic method used for further processing. The layer model avoids to pass parameters from the discretization methods to the algebraic methods. If a discretization scheme provides solution parameters to a certain kind of algebraic solution method, the software products for discretization and solution can be only used in this very special combination.

Once a new algebraic method is more apt to deal with the discrete problem, the specially determined parameters become obsolete or other parameters are required which implies additional implementation overhead. If a number of discretization schemes is thus bound to one solver interface, the use of another solver interface or even the use of an update of the solver interface with new interfaces might require an updating of all discretization methods.

Michael 2008-01-16