The Numerical Study On The Solution Landscape Of Singularly Perturbed Semilinear Elliptic Equations

In this paper,we are aimed to study the numerical algorithm for a class of singularly perturbed semilinear elliptic Neumann problems.By constructing the path diagram of solutions on the energy landscape,we explore the solution landscape of singularly perturbed semilinear elliptic problems from the per-spective of numerical calculation,that is,all solutions(including saddle points and extreme points of energy functional)and their relationships.First of all,the Morse indices of the trivial solutions of the singularly perturbed model problem,the bifurcation points of the trivial solution and the critical perturbation value₍(8)which determines the existence of the nontrivial positive solution,are introduced.Then the gentlest ascent dynamics of saddle points and the optimization-based shrinking dimer method are reviewed.Next,inspired by the work of J.Yin,L.Zhang and P.Zhang,based on the concept of solution landscape and two solution-path search methods,the saddle point dynamic equation is proposed for solving the general singularly perturbed semilinear elliptic problem in this paper.Moreover,an improved saddle point dynamics method for calculating the general Morse index saddle points is designed by combining the local grid,Newton acceleration and BB step size strategies.Finally,by using the above improved saddle point dynamic method and the downward search method,the solution landscape of the singularly per-turbed semilinear elliptic Neumann boundary problems for the Keller-Segal model with classical-structure and the Allen-Cahn model with classical(2-structure are computed,respectively.Meanwhile,for these two model problem-s with different fixed singular perturbation parameters,all the corresponding positive solutions searched from the root solutions as well as their relationships are obtained.As a result,the wonderful solution landscape of these two types of important equations is demonstrated.