The Super-convergence Method For Eigenvalue Problem With Discontinuous Coefficients

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 thesis, we consider eigenvalue problems with discontinuous coefficients. We combine spectral and finite element methods to construct a C₀ spectral element method. The method has strong ability to simulate the eigenvalue problem with discontinuous coefficients. It has super-geometric convergence rate. We provide a rigorous proof of this fact and our numerical tests support the theoretical result.This subject is recently studied by M.S.Min and D.Gottlieb[l8]. They reconstructed base function for the C₁ spectral element method, which results in a complex process. In contrast, construction for our C₀ spectral element method is much simpler. As a matter of fact, conventional spectral basis functions can be used directly, and still maintaining geometric convergence rate. Therefore, the C₀ spectral element method has advantage in practice over the C\ spectral element method.Main results of this thesis include:(1) For discontinuous eigenvalue problems Ql and Q2, we find all exact eigenvalues and eigenfunctions.(2) For the eigenvalue problem Ql with discontinuous coefficients and the Direchlet boundary condition, we use the traditional Lobatto-Gauss basis functions to construct a C₀ spectral element method. This construction is much simpler than the C₁ basis functions without sacrificing the accuracy. In addition, we are able to prove a super-geometric convergence. We give detailed analysis of our method, derived explicit form of the discrete matrix, establish the optimal error bounds, and present numerical examples.(3) For the eigenvalue problem Q2 with discontinuous coefficients and the periodic boundary condition, we study the Fourier-Galerkin method. Theoretical analysis has shown that the rate of convergence of this method is of order 3 for the eigenvalues and 2.5 for the eigenfunctions. These rates are confirmedby numerical tests. As a comparison, the Co spectral element method is much more accurate and efficient.