| This dissertation is mainly focused on study of the spectral methods for discontinuous problems arising in electromagnetism and medical imaging.; First, in electromagnetic waveguides, the Maxwell's equations can be cast as an eigenvalue problem. Of special interest are periodic structures of the domain which give rise to many interesting optical phenomenon in optical systems. When two different media are considered in the periodic structure, the eigenproblem will have a discontinuity in the dielectric coefficient function.; The eigenproblem can be reformulated in a variational form. When the Fourier-Galerkin and Fourier-Collocation methods are considered, the numerical solutions obtain only up to the third order accuracy for the eigenvalues and the eigenfunctions, due to the discontinuity in the dielectric coefficient of the eigenproblem. The convergence analysis for the Fourier methods is carried out by using the Minmax principle and the approximation theory on spectral methods.; Another approach to solve the discontinuous eigenproblem is splitting the domain into two subdomains and rewrite the eigenproblem into the multi-domain variational formulation. The Legendre-Galerkin, Legendre-Collocation, Legendre-Collocation Penalty methods are discussed. The exponential rate of convergence to the eigensolutions is confirmed both theoretically and numerically.; A common method, the Yee scheme (Finite Difference Time Domain method), as a direct simulation to solve the Maxwell's equation is discussed, specially on the instability of the Yee scheme for the "magic time step".; Second, a fast version of the Gegenbauer method is discussed in one and two dimensions. The computational cost and time comparisons with recurrence formula are presented. When the fast version of the Gegenbauer method is applied to a phantom in age reconstruction, assuming the location of discontinuities of the image and the Fourier coefficient data are known, it recovers the original image with superior quality compared to the exponential filters and Vandeven filters. The computational experiments are also performed for the Fourier coefficient from X-ray projections obtained by the direct Fourier method. |
