Legendre-tau Chebyshev collocation spectral element method for Maxwell's equations with material interfaces of two dimensional transverse magnetic mode

In this paper, a Legendre-tau Chebyshev collocation spectral element method is developed for solving Maxwell's equations with material interfaces in two dimensions. The transverse magnetic mode is considered mainly. The developed scheme treats the interface conditions in a way like the natural boundary condition based on a reasonable weak formulation, which makes the numerical solution retain the original physical properties. The domain is decomposed into some subdomains naturally at material interfaces. The electric and magnetic fields are approximated by using polynomial spaces of different degrees so that they can be solved separately in computation. Energy conservation and optimal error estimates are obtained. The fourth-order Runge-Kutta method is used in time advancing. Numerical experiments confirm that the spectral accuracy is achieved being not affected by the discontinuity of solutions. Compared with some related methods, the computational cost time of the scheme is shorter.