  In this paper, we develop an efficient spectral-Galerkin-type search
extension method (SGSEM) for finding multiple solutions to semilinear elliptic
boundary value problems. This method constructs effective initial data for
multiple solutions based on the linear combinations of some eigenfunctions of
the corresponding linear eigenvalue problem, and thus takes full advantage of
the traditional search extension method in constructing initials for multiple
solutions. Meanwhile, it possesses a low computational cost and high accuracy
due to the employment of an interpolated coefficient Legendre-Galerkin spectral
discretization. By applying the Schauder's fixed point theorem and other
technical strategies, the existence and spectral convergence of the numerical
solution corresponding to a specified true solution are rigorously proved. In
addition, the uniqueness of the numerical solution in a sufficiently small
neighborhood of each specified true solution is strictly verified. Numerical
results demonstrate the feasibility and efficiency of our algorithm and present
different types of multiple solutions.
