The Spectral Galerkin Method For Solving Multiple Solutions Of Semi-linear Elliptic Equations

In this paper, the theory and numerical methods for solving multiple solu-tions to a semi-linear elliptic Dirichlet problem are investigated, which expands the application regime of spectral Galerkin method. First of all, a search ex-tension method (SEM) based on spectral Galerkin method is designed. This method not only takes advantage of the traditional SEM in constructing good initial data for multiple solutions with characteristic basis functions, but al-so use the high accuracy and easy implementation properties of the spectral Galerkin method. At the same time, by taking the Lagrange basis functions as the experimental functions, it is much convenient to apply the interpolation coefficient method to improve calculation efficiency. Then, corresponding to any true solution, the existence, uniqueness and the error estimates for the nu-merical solution computed by the new algorithm are mathematically proved, which illustrates that the method presented in this paper can achieve the spec-tral convergence. Finally, a variety of multiple solutions for a cubic nonlinear model both in the one- and two-dimension cases are computed. The numeri-cal results show that the search extension method based on spectral Galerkin method is a algorithm of high accuracy and high efficiency in solving multiple solutions.