Finite Element Simulation Of Fractional Allen-Cahn Equation And Its Theoretical Analysis

As a typical phase field model,the Allen-Cahn equation of classical second order has been widely applied to many complex interface problems,such as vesicle membrane,solid nucleation and displacement process of two immiscible fluids.Therefore,it has important theoretical value and practical significance to establish an efficient numerical simulation method.However,field experiments and mathematical analysis show that the formation and development of a large number of diffusion interfaces depend on the long-range forces of various phase particles.The diffusion process of long range force can be characterized by fractional derivative operator rather than integer derivative operator.Therefore,it is necessary to simulate the fractional phase field.This paper focuses on the following fractional Allen-Cahn equation:where ?=(0,1),r ?(0,1),K>0 is the diffusion coefficient,the fractional Laplace operator(-?)r is defined as Riesz-type potentials on the whole space R.This paper consists of two parts:1.The backward-Euler-finite-element method for the fractional Allen-Cahn equation.The Galerkin variational form is established for the above fractional Allen-Cahn equation.The time derivative part is discrete by the backward Euler method,and the space derivative part is discrete by the finite element method.Thus,the finite element fully discrete scheme of fractional order Allen-Cahn equation is proposed.The contraction mapping principle is used to prove the solvability and energy stability of the finite element discrete format solution.A new energy equation is proposed under the new energy definition form,and the corresponding error estimate is obtained based on the discrete Gronwall inequality and the ellipse projection.The non-locality of the fractional Laplace operator causes the coefficient matrix of the finite element equation to be non-sparse,when we use the conjugate gradient algorithm to solve the finite element equation,the corresponding calculation and storage are large.Therefore,we use the linearization idea and skillfully combine the conjugate gradient algorithm with the fast Fourier transform and the special Toeplitiz structure of the non-sparse matrix,and design a fast conjugate gradient algorithm,which reduce the corre-sponding calculation and storage to O(MlogM)and O(M).The accuracy of the theoretical analysis is verified through numerical experiments,and the coarsening process of the interface is clearly characterized.2.SAV method of fractional Allen-Cahn equation.We introduce an auxiliary scalar to the above-mentioned fractional Allen-Cahn equation,and the time derivative part is discrete by the C-N method,thus,the SAV semi-discrete scheme of the fractional Allen-Cahn equation is proposed.According to the characteristics of the discrete format,the unconditional energy stability of the format is subtly proved,and the corresponding time error estimate is obtained by the discrete Gronwall inequality.The accuracy of theoretical analysis was verified by numerical experiments.