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From almost one century, it has been known that vector fields E, H, D, B, etc., in the Maxwell equations are just "proxies" for more fundamental objects, the differential forms e, h, d, b, etc., that when integrated on lines or surfaces, as the case may be, yield physically meaningful quantities such as emf's, mmf's, fluxes, etc. This viewpoint helps separate the "non-metric" part of the equations (Faraday and Ampère), fully covariant, from the "metric" one (the constitutive laws), with more restricted (Lorentz) covariance. The usefulness of this viewpoint in computational issues has been realized more recently, and will be the main topic addressed in this survey. It makes the association of degrees of freedom with mesh elements such as edges, facets, etc. (instead of nodes as in traditioanl finite element techniques), look natural, whereas the very notion of "edge element" seemed exotic twenty years ago. It explains why all numerical schemes treat Faraday and Ampère the same way, and only differ in the manner they discretize metric-dependent features, i.e., constitutive laws. What finite elements, finite volumes, and finite differences, have in common, is thus clearly seen. Moreover, this seems to be the right way to advance the "mimetic discretization" or "discrete differential calculus" research programs, which many dream about: a kind of functorial transformation of the partial differential equations of physics into discrete models, when space-time continuum is replaced by a discrete structure such as a lattice, a simplicial complex, etc. Though total fulfillment of this dream is still ahead, we already have something that engineers –especially programmers keen on object-oriented methods– should find valuable: A discretization toolkit, offering ready-to-use, natural "discrete" counterparts to virtually all "continuous" objects discernible in the equations, fields, differential operators, v x B force fields, Maxwell tensor, etc.
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