A Class Of Spectral Element Methods For The Steklov Eigenvalue Problem

In this thesis the author discusses the spectral element approximation with Legendre-Gauss-Lobatto (LGL) node basis for solving the Steklov eigenvalue problem. The Stekloveigenvalue problem has important physical background and wide applications especiallyin fluid. The spectral element methods are the numerical methods which combine withspectral methods and finite element methods. It has not only the characteristic of highaccuracy the spectral method has, but also the advantage of the finite element methodsthat can be suitable to complex boundary. In recent years, spectral element methodshave gradually received scholars’ attention. In order to study the spectral element ap-proximation for the Steklov eigenvalue problem, this paper establishes the a priori errorestimate and the residual type a posteriori error estimate by using the polynomial inverseestimates, the interpolation error estimate and the spectral approximation theory. Inaddition, the local error indicator is constituted with the weighted internal residual andthe weighted boundary residual and the reliability and effectiveness of the a posteriorierror estimator are analysed. At last, several numerical experiments on the square andL-shaped domain, which support our theory, are presented.