Finite Element Method For Time Fractional Multi-scale Diffusion Equation Based On Homogenization Theory

Time fractional partial differential equations are widely used in the modeling of semiconductor physics,turbulence and condensed matter physics,because they can well describe the characteristics of memory and heredity.Inhomogeneous scale phenomena exists widely in porous media and other problems.The time fractional In-homogeneous scale diffusion equation has complex mathematical characteristics such as the whole process correlation,singularity of time and Inhomogeneous scale,so its numerical solution is difficult.Generally,it needs very fine mesh generation and efficient numerical discretization scheme to obtain a better numerical solution.In this paper,the multi-scale homogenization method is studied for the time fractional Inhomogeneous scale diffusion equation with periodic coefficient.Firstly,the solution is asymptotically expanded with respect to the small scaleεby using the scale change.Based on the multi-scale homogenization method,the element control equation,the homogenization equation and the first-order approximate solution of the equation with respect toεare obtained.Then,because the first-order approximate solution has the oscillation problem aboutεat the spatial boundary,we obtain the modified first-order approximate solution by introducing the spatial cut-off function,and solve the boundary oscillation problem.Because the regularity of the initial value of the time fractional equation is generally low,and the multi-scale homogenization method has high requirements for the regularity of the initial value,we solve the prob-lem of high requirements for the smoothness of the initial value by introducing the time truncation function and constructing the error equation with the initial value of 0.For the solution and the modified first-order approximate solution,the error estimates under the norm of L²,H¹,and the error estimates about the norm of H^-1under the fractional derivative are given.In the fully discrete scheme of homogeniza-tion equation,the L¹scheme is used to discrete the fractional derivative in time,and the finite element method is used in space.Finally,a numerical example is given.For an example with smooth initial value conditions,the images of the numerical so-lutions of the element control equation and the homogenization equation are shown,and the errors under L² and H¹ are given.The numerical results show that the errors are satisfies the theoretical results.Furthermore,an example with nonsmooth initial value is considered,and the numerical solution and error are also given.This method can also obtain better numerical solutions under the condition of nonsmooth initial value.