Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

Normalized Goldstein-type local minimax method for finding multiple unstable solutions of semilinear elliptic PDEs

Pages: 147 – 174

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n1.a6

Authors

Wei Liu (Key Laboratory of Computing and Stochastic Mathematics, School of Mathematics and Statistics, Hunan Normal University, Changsha, China)

Ziqing Xie (Key Laboratory of Computing and Stochastic Mathematics, School of Mathematics and Statistics, Hunan Normal University, Changsha, China)

Wenfan Yi (Institute of Mathematics, School of Mathematics, Hunan University, Changsha, China)

Abstract

In this paper, we propose a normalized Goldstein-type local minimax method (NG-LMM) to seek for multiple minimax-type solutions. Inspired by the classical Goldstein line search rule in the optimization theory in $\mathbb{R}^m$, which is aimed to guarantee the global convergence of some descent algorithms, we introduce a normalized Goldstein-type search rule and combine it with the local minimax method to be suitable for finding multiple unstable solutions of semilinear elliptic PDEs both in numerical implementation and theoretical analysis. Compared with the normalized Armijo-type local minimax method (NA-LMM), which was first introduced in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3):865–885, 2002] and then modified in [Z.Q. Xie, Y.J. Yuan, and J. Zhou, SIAM J. Sci. Comput., 34(1):A395–A420, 2012], our approach can prevent the step-size from being too small automatically and then ensure that the iterations make reasonable progress by taking full advantage of two inequalities. The feasibility of the NG-LMM is verified strictly. Further, the global convergence of the NG-LMM is proven rigorously under a weak assumption that the peak selection is only continuous. Finally, it is implemented to solve several typical semilinear elliptic boundary value problems on square or dumbbell domains for multiple unstable solutions and the numerical results indicate that this approach performs well.

Keywords

semilinear elliptic PDEs, global convergence, multiple solutions, local minimax method, normalized Goldstein-type search rule

2010 Mathematics Subject Classification

35B38, 35J20, 65K15, 65N12

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This work was supported by NSFC (91430107, 11771138, 11171104) and the Construct Program of the Key Discipline in Hunan. Yi’s work was also partially supported by the Fundamental Research Funds for the Central Universities 531118010207 and the NSFC Grant 11901185.

Received 16 December 2019

Accepted 12 August 2020

Published 24 March 2021