Theoretical Analysis And Research Of Spectral Methods

Spectral methods are from the classic Ritz-Galerkin method. It is an important and effective numerical method for solving the partial differential equations. The biggest advantages of spectral method is called "infinite order convergence"and can be implemented fast Fourier calculation. These advantages of spectral methods have caused the concern of many scholars. Therefore, carrying out the research work about spectral method is of great theoretical significance.This paper introduces the spectral method associated with the basic theory and basic ideas. Afterward, the basic problems in the second category based on the framework of spectral methods are analyzed and summarized, the use of orthogonal Legendre polynomials as special test function to construct the spectral form proposed. In this paper, Helmholtz equation with the homogeneous boundary conditions is solved by using Legendre-Galerkin spectral method. This format is simple and easy to compute. At the same time, this paper carries out the error analysis and the corresponding numerical experiments. In the end, numerical experiments demonstrates that the Legendre polynomials as a special basis functions of Legendre-Galerkin method can maintain the numerical stability, the format have a practical significance.