The Research Of Numerical Methods For Solving Multiple Solutions Of Singularly Perturbed Semi-linear Elliptic Equations And Related Problems

In this paper, the theory and numerical methods for solving multiple solu-tions to a classical singularly perturbed Neumann problem, a general singularly perturbed Neumann problem and a semi-linear elliptic Dirichlet problem are investigated. A modified local minimax method (LMM) and its convergence analysis are proposed for the singularly perturbed multiple solution models. A search extension method (SEM) based on the Chebyshev spectral collocation method and its convergence analysis are displayed for the semi-linear ellip-tic Dirichlet problem. Some theoretical results for the singularly perturbed multiple solution models are also proved in this paper.Firstly, by relaxing the strict orthogonality requirement in selecting an initial guess for the traditional local minimax method(LMM) and introducing the local refinement strategy and other strategies, a modified LMM algorithm is designed in this paper. The sequences generated by the new algorithm are proved to converge to a new solution if the three conditions that the peak selection p is continuous, the separable condition and the energy functional is bounded below are all satisfied. The least energy solutions, boundary cor-ner multi-peak solutions, boundary non-corner multi-peak solutions, interior single-peak solutions and interior multi-peak solutions of the singularly per-turbed Neumann problems are systematically computed for the first time in the numerical examples.Next, motivated by the numerical results, the Morse indices of the trivial solutions at any value of ε, the bifurcation points of the trivial solutions and the critical values εc1and εc2, which determine the existence or nonexistence of a nontrivial positive solution and a nontrivial negative solution, are verified for the singularly perturbed models (1-1) and (4-1) respectively. The paper discusses some properties of the least energy solutions and sign-changing so-lutions and proposes a conjecture that the increasing orders of the energy and the maximum for the same kind of sign-changing solutions are p+1/p-1and1/p-1respectively, provided that ε is sufficiently large. All these theoretical results are proved by a large number of numerical results. Finally, a SEM based on Chebyshev spectral collocation method is de-veloped for a semi-linear elliptic Dirichlet problem in this paper. Under the basic assumption (5-18), we analyze the convergence results of this method and show that the Hw1and Lw2errors between the numerical solutions by solv-ing model equation (1-3) with the Chebyshev spectral collocation method and the corresponding true solutions are O(N1-σ) and O(N-σ) respectively, which reflect the full orders of spectral collocation method. Furthermore, when N is sufficiently large, the numerical solution μN which sufficiently approaches a true solution u is unique. The numerical results state that the SEM based on Chebyshev spectral collocation method is more efficient than the traditional SEM based on finite element discretization and two-grid method.