A Class Of Barzilai-Borwein Type Local Minimax Algorithm For Computing Multiple Solutions Of Semilinear Elliptic Equations

In this paper,we study the multi-solution theory and numerical algorithm for a class of semilinear elliptic partial differential equations.Because of the difficulty of nonlinearity of the model problem,instability and multiplicity of the solutions,it is extremely challenging to design a stable,efficient and convergent numerical algorithm.At present,many effective numerical algorithms have been successfully applied to the multi-solution problem,such as the mountain path method,the high-linking method,the local minimax method and search extension method.Based on LMM algorithm,this paper proposes a class of Barzilai-Borwein type LMM algorithm for model problems.The core idea of the algorithm is to construct the Barzilai-Borwein type step size and a class of nonmonotonic search strategy for solving the outer layer local minimum and maximum problem of LMM algorithm,and analyze the feasibility and convergence of the Barzilai-Borwein type local minimax algorithm,and this algorithm to solve several unstable solutions of the Lane-Emden equation,H?enon equation and the nonlinear Schr¨odinger equation,and obtain abundant numerical results.Compared with the traditional LMM algorithm,the results show that the algorithm has a faster convergence rate.