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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. [ABSTRACT FROM AUTHOR]

In the paper, an operator splitting Legendre-tau spectral method is proposed to solve Maxwell's equations with nonlinear conductivity in two dimensions. By the implicit–explicit numerical scheme, the linear and nonlinear parts are treated in different ways. The linear terms are treated by the Legendre-tau spectral method implicitly, and the nonlinear terms are treated by some collocation methods explicitly. The scheme uses polynomial spaces of different degrees to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. The splitting leap frog Crank–Nicolson method is applied in time discretization, which is a three-level scheme with the nonlinear term being treated explicitly in intermediate level. With the application of inverse inequality, for the Legendre interpolation operator, optimal L 2 -error estimates are achieved for the proposed scheme. In numerical examples, the nonlinear terms are computed at Chebyshev–Gauss–Lobatto points by the fast Legendre transform. Numerical results validate the efficiency and spectral accuracy of the scheme. [ABSTRACT FROM AUTHOR]

A multidomain Legendre–Chebyshev spectral method is developed for solving two dimensional Maxwell's equations in inhomogeneous media with discontinuous electromagnetic waves. The method keeps spectral accuracy being not affected by the discontinuity of solutions. We construct a reasonable weak form which deals with the interface conditions similar to the natural boundary condition. Polynomial spaces of different degrees are used to approximate the electric and magnetic fields so that they can be decoupled in computation, and the optimal error estimate is obtained which improves the previous results. Numerical examples confirm the higher accuracy compared with other related method. [ABSTRACT FROM AUTHOR]

In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. The approximation results in Legendre norm rather than in the Chebyshev weighted norm are given, which play a fundamental role in numerical analysis of the Legendre–Chebyshev spectral method. These results are also useful in Clenshaw–Curtis quadrature which is based on sampling the integrand at Chebyshev points. [ABSTRACT FROM AUTHOR]