Annals of Mathematical Sciences and Applications

Volume 4 (2019)

Number 1

Matrix representations of discrete differential operators and operations in electromagnetism

Pages: 55 – 79



Tsung-Ming Huang (Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan)

Wen-Wei Lin (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Weichung Wang (Institute of Applied Mathematical Sciences, National Taiwan University, Taipei, Taiwan)


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 $\nabla \times, \nabla \cdot, \nabla, \nabla^2, \nabla (\nabla \cdot)$, 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.


Maxwell’s equations, Yee’s discretization scheme, matrix representation, curl, divergence, gradient, periodic structures, simple cubic lattice, face centered cubic lattice

This work is partially supported by the Ministry of Science and Technology, the Taida Institute of Mathematical Sciences, the Center for Advanced Study in Theoretical Sciences, and the National Center for Theoretical Sciences in Taiwan, and the ST Yau Center at NCTU.

Received 18 September 2018

Published 26 February 2019