Matrix representations of discrete differential operators and operations in electromagnetism

Metamaterials with periodic structures are building blocks of various photonic and electronic materials. Numerical solutions of three dimensional Maxwell's equations, play an important role in exploring and design these novel artificial materials. Yee's finite difference scheme has been widely used to discretize the Maxwell equations. However, studies of Yee's scheme from the viewpoints of matrix computation remain sparse. To fill the gap, we derive the explicit matrix representations of the differential operators del x, del., del, del(2), del(del.), and prove that they satisfy some identities analogous to their continuous counterparts. These matrix representations inspire us to develop efficient eigensolvers of Maxwell's equations and help to show the divergence free constraints hold in Yee's scheme.