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This chapter presents a set of conceptual tools for the formulation of physical field problems in discrete terms. These tools allow the representation of the geometry and of the fields in discrete terms and those of chain and cochain. Moreover, they allow bridging the gap between the continuous and the discrete concept of field by means of the idea of limit systems. 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 privileged discrete operator—the coboundary operator—exists for the representation of topological laws. The chapter discusses a reference discretization strategy that complies with these concepts. It is based on the idea of topological time stepping for time-dependent equations that operates on global quantities and derives from the application of the coboundary operator in space–time.

@Article{ Mattiussi_2000aa, abstract = {This chapter presents a set of conceptual tools for the formulation of physical field problems in discrete terms. These tools allow the representation of the geometry and of the fields in discrete terms and those of chain and cochain. Moreover, they allow bridging the gap between the continuous and the discrete concept of field by means of the idea of limit systems. 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 privileged discrete operator—the coboundary operator—exists for the representation of topological laws. The chapter discusses a reference discretization strategy that complies with these concepts. It is based on the idea of topological time stepping for time-dependent equations that operates on global quantities and derives from the application of the coboundary operator in space–time.}, author = {Mattiussi, Claudio}, doi = {10.1016/S1076-5670(00)80012-9}, file = {Mattiussi_2000aa.pdf}, journal = {Advances in Imaging and Electron Physics}, keywords = {fem,fit,field,fdtd,physics,fvm}, langid = {english}, pages = {1--146}, title = {The Finite Volume, Finite Difference, and Finite Elements Methods as Numerical Methods for Physical Field Problems}, url = {http://infoscience.epfl.ch/record/63908}, volume = {113}, year = {2000}, shortjournal = {Adv. Imag. Electron. Phys.} }

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