The initial intention on which the discrete calculations of Chapter 2 is based is a common framework for the specification of different discretization schemes. It could be shown in Chapter 3 that the commonly used differential equations can be discretized using the provided framework of functional expressions.
The basic features of the discrete formalism is the use of accumulation operations. As a consequence, not only local formulae can be evaluated but values within the neighborhood of a given topological element can be accessed and used for the evaluation of the differential operator in that topological element. While in other environments the use of traversal operations is strictly reduced to a set of base operations, the provided framework relies on the underlying topological environment which provides different kinds of traversal operations as well as topological functions.
The main advantage is that discrete expressions can be directly transformed into code which eases the implementation and reduces the possibility of flaws and oversights. Furthermore, the absence of iteration variables within the traversal operations makes the code easier to adapt and maintain.