A Numerical Method For The Analysis Of Stochastic Allen-Cahn Equation

In this paper,the numerical schemes are presented for stochastic Allen-Cahn equation from two different perspectives.On the one side,the semi-discrete scheme is obtained by spatial discretization with local discontinuous finite element method.At the same time,the existence and uniqueness of the numerical solution is proved,the stability of the numerical scheme is obtained and the error is analyzed.On the other hand,the approximation of the strong solution based on the local discontinuous finite element method is obtained,and two different numerical methods are used to approximate the nonlinear term.Moreover,the error analysis is carried out and the optimal error estimation is obtained.Finally,the theoretical results are verified by numerical experiments.This paper consists of the following five chapters.In chapter 1,we firstly introduce the background,the significance and the research progress of the stochastic partial differential equations.In addition,we review the development process of the local discontinuous Galerkin method.Moreover,it points out the main content and the structure arrangement of this paper.In chapter 2,the theoretical basis of some stochastic differential equations and the preparatory knowledge are introduced.In chapter 3,stochastic Allen-Cahn equation is discretized by using local discontinuous finite element method and the semi-discrete scheme is obtained.Chapter 4 is devoted to study the full-discrete scheme of stochastic Allen-Cahn equation and the convergence of the strong solution is analyzed.In chapter 5,the theoretical results are verified by numerical experiments.Finally,a summary of the research content is made and we look forward to the developments of the future.