Numerical Solution Of Allen-Cahn Equation Driven By Gradient Mutiplicative Noise

Allen-Cahn equation is of great significance in the study of materials science,it has be-come a fundamental equation and component of general moving interface problems in field methodology.However,in many applications of mean curvature flow,there may be uncertain-ties caused by thermal expansion of materials or inherent instability of materials.Therefore,in this paper,an Allen-Cahn equation with gradient multiplicative noise is obtained by adding the noise term into the determined mean curvature flow according to the geometric law.The Allen-Cahn equation in this paper contains gradient-type multiplicative noise and belongs to the strongest noise form of the second-order quasilinear partial differential equation.Compared with deterministic equation,it is more extensive,applicable and innovative in real life.Firstly,the strong solutions of the equation are regularized.Because of the existence of non-linear term in the model,the global Lipschitz condition is not satisfied,so the error analysis becomes very challenging.In this paper,by introducing an auxiliary approximation process,the error terms considered are decomposed appropriately.The finite element method is used to semi-discretize the equation,Then,the implicit finite difference scheme and the finite element method are used to discretize the equation,then the convergence error estimates are obtained.