The Eigenvalue Problem With A Piecewise Constant Coefficient Using Spectral Method

The calculation of eigenvalue problem is one of the basic topics of scientific computing. It has a wide range of applications in science, engineering, economy, and management.In this paper, we consider a second order model eigenvalue problem with discontinuous coefficients. It is motivated by nonlinear optics. The spectral methods when applied to problems in nonlinear optics are extremely fast, and if the problem is smooth they provide high accuracy. However, when different media is considered, the coefficients are only piecewisly smooth and the accuracy lost. We split the domain into two parts at the discontinuity of the coefficient function, define basic functions in each part independently, and define a special basic function to connect these two parts at last. The discrete multidomain variational formulation formed the key point of the paper: Legendre-Galerkin spectral method. Numerical results show that by splitting the domain the spectral rate of the approximation is obtained for the Legendre-Galerkin spectral method. The minmax principle is used for the convergence analysis.The Chebyshev collocation method with pointwise evaluation of the discontinuous coefficients are also considered for the eigenvalue problem. In this case we obtain only the second-order accuracy. This could be improved by using a truncated Chebyshev series of the coefficients. Numerical results are presented to show the convergence rates for the different spectral schemes.