A minimax algorithm based on Newton's method and an application for finding multiple solutions of p-area problems

In this paper, we present a numerical scheme which is an improved minimax method proposed by Yao-Zhou's study along with Newton's method for finding the multiple solutions of p-area problems. It is combined with Newton's method since the local minimax method stagnates close to a critical point of the functional whereas Newton's method converges rapidly close to a critical point. We discuss the theoretical convergence result of this proposed hybrid scheme and the convergence of the finite element based solutions as well. Further, we carry out instability analysis of the unstable solutions through their local minimax type characterizations, where the instability behavior of a solution can be analyzed before calculating the solution. This analysis also provides ways to numerically compute the Morse index of the solutions. Finally, numerical experiments are performed to demonstrate the algorithm and the theoretical results.

Automated summary

In this paper, a numerical scheme combining an improved minimax method and Newton's method is presented for finding multiple solutions of p-area problems. The theoretical convergence result, the convergence of finite element based solutions, and the instability analysis of unstable solutions through local minimax characterizations are discussed. Numerical experiments are also conducted to demonstrate the algorithm and theoretical results.