Accelerating Augmented Partial Newton Method For Finding Multiple Solutions Of Nonlinear PDEs

In this paper,we mainly used the Augmented Partial Newton Method with Newton Acceleration(AAPNM)for calculating multiple solutions of the nonlinear partial differential equations.The multiplicity and the instability of the solutions to the nonlinear partial differential equations bring a lot of difficulties both to the design of the numerical methods and mathematical analysis.How to design a stable numerical algorithm foe unstable solutions,reduce the dependence of an initial value,and ensure that each solution obtained is new,are very important and extremely challenging.A new algorithm has been put forward in this article:Accelerating Aug-mented Partial Newton Method(AAPNM).Its main idea is to obtain a rough approximate solution by using the the method of Augmented Partial Newton Method(APNM),and then using the classical Newton mehtod to accelerate the convergence.This method not only inherits the large convergence of the Augmented Partial Newton method,but also guarantees that the solution of each calculation must be a new solution,further on,it retains the importan-t characteristic of the local quadratic convergence rate of classical Newton method,and realizes the perfect combination of APNM and Newton method.In this paper,the AAPNM is used to calculate the multiple solutions problem of nonlinear partial differential equations with non-variational struc-ture and the multiple solutions problem of singularly perturbed semi-linear elliptic equations with Neumann boundary conditions,and some interesting numerical results are obtained.