The finite volume, finite element, and finite difference methods as numerical methods for physical field problems

A set of conceptual tools for the formulation of physical field problems in discrete terms is presented. These tools allow the representation of the geometry and of the fields in discrete terms, using the concept of oriented cell-complex, and those of chain and cochain. Moreover they allow us to bridge the gap between the continuous and the discrete concept of field by means of the idea of limit systems. The analysis of the structure of physical field theories is based on these tools. This analysis unveils the importance of thinking of physical quantities as associated with space-time oriented geometric objects. Moreover, this analysis exposes the distinction of topological laws from constitutive relations showing their different behavior from the point of view of their discretizability. A reference discretization strategy, which complies with these concepts, has also been presented. It is based on the idea of topological time-stepping for time-dependent equations, which operates on global quantities and derives from the application of the coboundary operator in space-time.