Convergence results of a local minimax method for finding multiple critical points

In [Y. Li and J. Zhou, SIAM J. Sci. Comput., 23 (2001), pp. 840-865], a new local minimax method that characterizes a saddle point as a solution to a local minimax problem was established. Based on the local characterization, a numerical minimax algorithm was designed for finding multiple saddle points. Numerical computations of many examples in semilinear elliptic PDE were successfully carried out to solve for multiple solutions. One of the important issues which remains unsolved is the convergence of the numerical minimax method. In this paper, first Step 5 in the algorithm is modified with the design of a new stepsize rule that is easier to implement practically and with which convergence results of the numerical minimax method are established for isolated and nonisolated critical points. The convergence results show that the algorithm indeed exceeds the scope of a minimax principle. In the last section, numerical multiple solutions to the Henon equation and a sublinear elliptic equation subject to zero Dirichlet boundary condition are presented to show their numerical convergence and profiles.