Least-squared Mixed Finite Element Method And Theoretical Analysis For Time-space Fractional Diffusion Problems

In recent years,many anomalous diffusion phenomena,for examples,the plume flow with long tail in the sediment transport in an estuary,the dispersive transport of electrons in operation of photocopiers and laser printers,were reported in such practical fields as non-Fick diffusion,chaotic dynamics,viscoelastic mechanics,etc..These precesses were commonly modeled by second-order diffusion equation before.Numerous experiments and field tests have showed that the fractional-order dif-ferential equations can provide more adequate descriptions than the integer-order differential equations do.Since it is rarely available to find its closed form for the fractional-order differential equations through Fourier transform.Laplace transform and other techniques,one turn his attention to numerical methods,such as finite element and finite difference methods,for their approximation.To accordance with the engineering needs,an ideal numerical method should recognize the unknown function as well as its diffusion flux which is.at least,as important as the unknown in engineering.This was realized by introducing the flux u=-Dp as an intermedi-ate variable and and then formulating a mixed finite element procedure under the assumption that the related two function spaces should satisfy the LBB conditionIn this paper,we still introduce the diffusion flux u=-Dp as an intermediate variable and apply the least-squared technique to reformulate the fractional-order diffusion equation into a mixed variational form,and thus propose a least-squared mixed finite element method without the LBB assumption for the related two func-tion spaces.In the first part of this paper,we discuss the following time-dependent frac-tional diffusion equation of order 2-? characterized by one-side Riemann-Liouville fractional derivatives.Here,?=(0,1),0<?<1,f?L2(Q)is the source or sink term;?/?t and D=d/dx are the first-order derivative operators for time and space.0Dx refer to the left Riemann-Liouville fractional-order integral operators of order 1-? defined below by(2.2.3).In this part,we combine the idea of Chen and Wang in[6]with the least-squared technique and propose a least-squared mixed finite element method,by introducing the diffusion flux u=-Dp as intermediate variable,approximating the time deriva-tive by the Euler schemem and reformulating the fractional-order diffusion equation into a least-squared mixed variational form.We prove the solvability of the mixed finite element solution by applying the Garding inequality or the convexity of the least-squared functional and by establishing the equivalence between the minimiza-tion of the least-squared functional and the mixed variational form.Numerical tests are provided.The advantages of the proposed method are that the LBB condition can not be required for the finite element spaces and the unknown as well as the diffusion flux can be calculated with high accuracy.Noticing that a large number of anomalous diffusion or non-Fick transport processes are super-diffusive,i.e.the derivatives of time and space are fractional order,we shall study the following space-time fractional order diffusion problems in the second part of this paper,Here 0D?tp(x,t)represents the Caputo fractional-order derivative of order ?,0<?<1.We introduce the diffusion flux u=-Dp as the intermediate variable again and discrete the Caputo derivative by the L1-scheme,then reformulate the space-time fractional order diffusion problem into a least-squared mixed weak form.From this,we propose a least-squared mixed finite element method.We prove the solvability of the mixed finite element solution by applying the Carding inequality or the convexity of the least-squared functional and by establishing the equivalence between the minimization of the least-squared functional and the mixed variational form.Numerical tests are provided.As mentioned in the first part,the least-squared mixed finite element method combined with the L1-scheme can approximate the unknown and the diffusion flux well without the LBB constraints for the finite element spaces.