Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems

Abstract

We study computational theory and numerical methods for finding multiple unstable solutions (saddle points) for two types of nonlinear variational functionals. The first type consists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated by quasilinear elliptic problems from physical applications, where energy functionals are at most lower semi-continuous with blow-up singularities in the whole space and G-differntiable in a subspace, and mathematical results and numerical methods for C1 or nonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish a new mathematical frame-work for a local minimax method and its numerical implementation for finding multiple G-saddle points with a new strong-weak topology approach. Numerical implementation in a weak form of the algorithm is presented. Numerical examples are carried out to illustrate the method. The second type consists of C^1 W-type (defocused) problems. In many applications, finding saddles for W-type functionals is desirable, but no mathematically validated numerical method for finding multiple solutions exists in literature so far. In this dissertation, a new mathematical numerical method called a local minmaxmin method (LMMM) is proposed and numerical examples are carried out to illustrate the efficiency of this method. We also establish computational theory and present the convergence results of LMMM under much weaker conditions. Furthermore, we study this algorithm in depth for a typical W-type problem and analyze the instability performances of saddles by LMMM as well.