The question of the definition of an exact counterpart of Poynting's Theorem for the Maxwell equations on a simplicial complex or cell complex is considered. The cell complex approach is purely discrete and the discrete Maxwell equations in this framework are exact, rather than approximations to some order in a small mesh parameter of the continuum counterparts. The FDTD method is a direct example of Maxwell's equations on a cell complex with discrete time-stepping. An energy density is defined, along with a suitable candidate for a Poynting flux, and it is required that exact conservation hold over arbitrary closed surfaces in the complex.
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