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

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Title: Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems
Authors: Liu, Wei, Xie, Ziqing, Yuan, Yongjun
Source: SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-1431
Publication Year: 2023
Collection: Computer Science
Mathematics
Subject Terms: Mathematics - Numerical Analysis, 35J25, 65N35, 65H10, 47H10
More Details: 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.
Comment: 23 pages, 7 figures; Chinese version of this paper is published in SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-1431
Document Type: Working Paper
DOI: 10.1360/SCM-2019-0357
Access URL: http://arxiv.org/abs/2308.06529
Accession Number: edsarx.2308.06529
Database: arXiv
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  Data: Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems
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  Data: <searchLink fieldCode="AR" term="%22Liu%2C+Wei%22">Liu, Wei</searchLink><br /><searchLink fieldCode="AR" term="%22Xie%2C+Ziqing%22">Xie, Ziqing</searchLink><br /><searchLink fieldCode="AR" term="%22Yuan%2C+Yongjun%22">Yuan, Yongjun</searchLink>
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  Data: SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-1431
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  Data: 2023
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  Data: 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.<br />Comment: 23 pages, 7 figures; Chinese version of this paper is published in SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-1431
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