A Local Minimax Method Using the Generalized Nehari Manifold for Finding Differential Saddles

Abstract

In order to find the first few unconstrained saddles of functionals with different types of variational structures, a new local minimax method (LMM), based on a dynamics of points on virtual geometric objects such as curves, surfaces, etc., is developed. Algorithm stability and convergence are mathematically verified. The new algorithm is tested on several benchmark examples commonly used in the literature to show its stability and efficiency, then it is applied to numerically compute saddles of a semilinear elliptic PDE of both M-type (focusing) and W-type (defocusing). The Newton’s method will also be investigated and used to accelerate the local convergence and increase the accuracy.

The Nehari manifold is used in the algorithm to satisfy a crucial condition for convergence. The numerical computation is also accelerated and a comparison of computation speed between using the Nehari manifold and quadratic geometric objects on the same semilinear elliptic PDEs is given, then a mixed M and W type case is solved by the LMM with the Nehari manifold.

To solve the indefinite M-type problems, the generalized Nehari manifold is introduced in detail, and a generalized dynamic system of points on it is given. The corresponding LMM with a correction technique is also justified and a convergence analysis is presented, then it is tested on an indefinite M-type case. A numerical investigation of bifurcation for an indefinite problem will be given to provide numerical evidence for PDE analysts for future study