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.

翻译：本文发展了一种高效的谱-伽辽金型搜索扩展方法（SGSEM），用于求解半线性椭圆边值问题的多解问题。该方法基于相应线性特征值问题中若干特征函数的线性组合，为多解构造了有效的初始数据，从而充分发挥了传统搜索扩展方法在构造多解初值方面的优势。同时，由于采用了插值系数型Legendre-Galerkin谱离散格式，该方法具有低计算成本和高精度的特点。通过应用Schauder不动点定理及其他技术策略，严格证明了对应于指定真解数值解的存在性与谱收敛性。此外，还严格验证了每个指定真解足够小邻域内数值解的唯一性。数值结果展示了该算法的可行性与高效性，并呈现了不同类型的多解现象。