Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems

TL;DR

This paper introduces a spectral-Galerkin-type search extension method (SGSEM) for efficiently finding multiple solutions to semilinear elliptic boundary value problems, with proven convergence and verified uniqueness.

Contribution

The paper develops a novel SGSEM that constructs initial guesses from eigenfunctions, offering high accuracy and low computational cost for multiple solutions.

Findings

Proves existence and spectral convergence of solutions.

Demonstrates the method's efficiency through numerical experiments.

Verifies the uniqueness of solutions in small neighborhoods.

Abstract

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.