On Finding Multiple Solutions to a Singularly Perturbed Neumann Problem

Abstract

Abstract. In this paper, in order to numerically solve for multiple positive solutions to a singularly perturbed Neumann boundary value problem in mathematical biology and other applications, a local minimax method is modified with new local mesh refinement and other strategies. Algorithm convergence and other related properties are verified. Motivated by the numerical algorithm and convinced by the numerical results, a Morse index approach is used to identify the Morse index of the root solution u 1 ε = 1 at any perturbation value, its bifurcation points and then the critical perturbation value. Many interesting numerical solutions are computed for the first time and displayed with their contours and mesh profiles to illustrate the theory and method.

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

“…Motivated by classical minimax theorems in the critical point theory (see, e.g., [21] and references therein) and numerical researches of Choi-McKenna [5], Ding-Costa-Chen [8] and Chen-Zhou-Ni [4], Li and Zhou proposed a local minimax method (LMM) for various high-MI saddle points based on a local minimax characterization of them [16]. Then in [27], Xie et al modified the LMM with a significant relaxation on the domain of the local peak selection, which is a crucial notion for the LMM and will be illustrated in details later. According to [16,27], the LMM grasps a saddle point with MI = n (n ∈ N + ) by dealing with a two-level local minimax problem as min v∈S H max w∈ [L,v] E(w), (1.2) where S H = {v ∈ H : v = 1} is the unit sphere with • the norm in H, L ⊂ H is a given (n − 1)-dimensional closed subspace usually constructed based on some known or previously found critical points, and [L, v] = {tv + w L : t ≥ 0, w L ∈ L} denotes a closed half subspace.…”

Spike-Layer Simulation for Steady-State Coupled Schrödinger Equations

“…Refs. [3,11,15]. For the Dirichlet boundary condition, Chen et al [3] introduced the Mountain Pass Algorithm (MPA) to obtain the interior single-peak solution on an elliptic domain when ǫ ≥ 10 −1 .…”

“…Ávilaa et al [15] extended the MPA idea to consider singularly perturbed problems, and developed a finite element approach combined with steepest descent calculations to obtain some interior single-peak solutions on a circular domain when ǫ ≥ 10 −6 , or on a dumbbell domain when ǫ ≥ 10 −5 . For the Neumann boundary condition, Xie et al [11] developed a modified Local Maximum Method (LMM) that produced boundary single-peak solutions, boundary multiple-peak solutions, and interior single-peak solutions.…”

Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs

“…However, from various recent works, e.g. [107], one does expect that it is possible to modify the approach to the Caginalp system, as well as other variational problems to find starting solutions for continuation; this approach will be considered in future work. For the Caginalp system we find an interesting bifurcation diagram with potential applications to tune parameter values to access different parts of the so-called mushy regime between melting and solidification.…”