A Least-squared Mixed Finite Element Method For 2-D Fractional Diffusion Equations

In this paper,we discuss the following fractional diffusion equation in two dimensional space,where ? =[0,1]×[0,1],0<?<1,p(x,y)is diffusion concentration,f(x,y)is source term and the diffusion coefficient of the medium is assumed to be constant.?and ?1-?represent the gradient operator and the fractional divergence operator of order 1-?,respectively.In order to meet the needs of engineering practice,an ideal numerical simulation method should simultaneously make a high-precision approximation to the unknown function and its flux.However,we find that the numerical method based on the difference framework can only simulate the unknown function and most of the nu-merical methods based on the finite element framework are limited to the discussion of one-dimensional fractional problems.There are few numerical methods and cor-responding numerical analysis theories that are widely applied to two-dimensional fractional diffusion problems.In this paper,by using the idea of operator splitting and introducing the diffu?sion flux u =-?P,we decompose the two-dimensional fractional diffusion equation into a system composed of two low-order equations.Then,we apply the least squared technique to establish the minimum problem,and obtain the mixed variational for-m to induce the corresponding mixed finite element scheme.We further prove the equivalence between the the variational form and the minimum problem.In order to prove the solvability of the variational form,we need to select the appropriate space and use Lax-Milgram lemma.Here,we choose Sobolev space H01(?)for the unknown p?which can ensure the solvability of p due to the equivalence of the norm and semi-norm in space H01(?).For the diffusion flux u,the fractional divergence space H1-?(div;?)can not be selected directly to be its admissible space since the norm and semi-norm are not equivalent in Hl-?(div;?).We turn to the fractional quotient space H1-?(div;?)/Ker{?1-?·} for the admissible space of u.This per-mits us to apply the Lax-Milgram lemma to prove the solvability of u.Then,we choose the lowest order Raviart-Thomas finite element space and the bilinear finite element space to propose the least squared mixed finite element scheme,and prove the existence and uniqueness of discrete solutions.Finally,numerical exper-iments are carried out to demonstrate the effectiveness of the least squared mixed finite element scheme.Because of the non-locality of fractional derivative operator,the coefficient matrix of the mixed finite element scheme is not sparse,which brings great trouble to the matrix computation.To solve this problem,we use the idea of blocking matrices and the properties of the fractional divergence operator to decompose the coefficient matrix into four blocked matrices with symmetry or identity property.This will reduce the cost of those entries computation for the coefficient matrix.