A self-contained electromagnetic theory is developed on a simplicial lattice. Instead of dealing with vectorial field, discrete exterior calculus (DEC) studies the discrete differential forms of electric and magnetic fields, and circumcenter dual is adopted to achieve diagonal Hodge star operators. In this paper, Gauss' theorem and Stokes' theorem are shown to be satisfied inherently within DEC. Many other electromagnetic theorems, such as Huygens' principle, reciprocity theorem, and Poynting's theorem, can also be derived on this simplicial lattice consistently with an appropriate definition of wedge product between cochains. The preservation of these theorems guarantees that this treatment of Maxwell's equations will not lead to spurious solutions.
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