The Mixed Finite Element Method And Theoretical Analysis Of The Fractional Diffusion Equation With Variable Coefficients

Diffusion,a process that caused by random movements of particles during the trans-formation of a system from non-homogeneous disequilibrium state to homogeneous equi-librium state,exists widely in industrial production and social economy.In 1822,Fourier,a French mathematician and physicist,deduced the diffusion equation of heat-heat con-duction equation,which was the first time to describe the diffusion process by integer differential equation;In 1855,Fick,a German physiologist,set up the diffusion equa-tion by applying Fourier s heat conduction equation when studying how nutrients travel through membranes in living organisms;In 1905,Einstein derived the diffusion equation from first principle as part of his work on Brownian motion.These second-order diffu?sion models are based on a common assumption that the displacement of micro-particles obeys the classical Gauss distribution,that is,the existence of a mean free path and theexistence of a mean waiting time taken to perform a jump.However,it was found that there are a large number of phenomena in nature,called"abnormal diffusion",which do not satisfy the above assumption and can't be modeled adequately and properly by integer order diffusion equation.For example,turbulencein meteorological and aerospace[29,107],plumes with long tails during the transport of sediment in estuaries,and the way that photocopier and the laser printer.A large number of experiments and data show that fractional diffusion equation can capture the"anomalous diffusion" and describe the anomalous phenomenon accurately.Therefore,the research on mathematical theory and numerical simulation for fractional diffusion equation has a great significance,not only to comprehensively understand the mechanism of diffusion process,but also to provide decision-making basis for practical engineering practice,and has become a hot research area in the field of engineering technology and mathematics.The structure of solution for fraction diffusion equation,involving fractional op-erator which is different from integer operator and is a nonlocal operator,is complex,so only a few fractional diffusion equations can obtain their analytical solutions,and the analytical solutions are expressed by special functions.It is not simple to express these special functions numerically,so the study on numerical solutions is particularly important.Series approximation methods,including variational iteration method,Adomian decomposition method and so on,are the numerical methods used to solve fractionaldiffusion in early times.However,although the numerical solution can converge to the exact solution at a faster speed,the calculation is generally very complicated because it involves the fractional derivatives of known functions.Especially in engineering ap-plications,in order to obtain higher accuracy,the number of terms of series must be increased,which undoubtedly increases the computational complexity,and it needs to be derived by symbolic calculation,which causes computers to have poor realizability and cannot meet the needs of simulation of complex engineering problems.Therefore,calculation methods which are simple,efficient and easy to be realized by computers,such as finite difference and finite element,have attracted considerable attentions in practical applications.Finite difference method is to approximate Riemann-Liouville fractional deriva-tive by using shifted Grunwald-Letnikov technique,based on the equivalence between Riemann-Liouville fractional derivative and Grunwald-Letnikov fractional derivative un-der certain conditions.It is widely used by scholars because of its simplicity,flexibility,versatility and easy programming.Based on this method,some other numerical methods have been developed,such as spectral methods[58,77,79,86,98,150,151],,multi-grid method[101],collocation method[105],finite volume method[145]and so on.Galerkin finite element method is also applied to construct numerical approximation for fraction diffusion equation.It relaxes the requirement for smoothness of solutions by using functional analysis as a tool,and incorporates analytical solutions and finite ele-ment solutions into a complete functional space so as to use the abundant mathematical tools to carry out numerical analysis.However,it needs some prior conditions such as well-posedness and regularity of solutions of differential equations.The two properties of fractional differential operators,that Fractional differential operator is a non-local operator and its adjoint operator is not the negative of itself,make the structure and properties of the solutions,such as existence and regularity,not as clear as those of classical diffusion equation and so bring essential difficulties for applying Galerkin-finite element technique to numerically simulate fractional diffusion equation.Therefore,it is reasonable to develop its Galerkin variational formulation for predicting its mathemati-cal properties and also inducing an easily-computed finite element procedure with high convergence accuracy.For constant coefficient fractional diffusion equation,Ervin and Roop[38,39,41]pre-sented a theoretical framework for the Galerkin finite element approximation to fractional diffusion equation and proved an optimal-order error estimate for the corresponding fi-nite element method assuming that the true solution of the problem has the requiredregularity.Since then,the discontinuous Galerkin method 32,33],the Petrov-Galerkin method[64,65,134,136]and so on,have emerged for fractional diffusion equation with a con?stant diffusion coefncient.In 2016,Chen and Wang[20]proposed a saddle-point framework and developed a locally-conservative expanded mixed finite element method by introducing two interme-diate variables q = Dp and u =-K0Ix?qAlthough a rigorous numerical analysis theory was established,the convergence rates for the unknown u and p were not optimal with respect to regularity requirement for sufficiently smooth right term f.This is attributed to the appearance of the weak singular term x1-?[64],which makes the regularity of the exact solution p only belong to H1-?+?(?),q belong to H?-?(?)and so the convergence rates are heavily destroyed to ?-?/2,??[?/2,1/2).In order to overcome the difficulties caused by the low regularity of solutions,many methods have emerged:refined the mese[60]at the neighbor of x =0 so as to obtain more accurate numerical so-lution,which may heavily increase the computing time;transformed variable method[63],singularity reconstruction strategy[66],and Petrov-Galerkin spectral method[86]to cap-ture this singular term.However,these methods couldn't extend to variable-coefficient diffusion equations.For variable-coefficient counterpart,the structure and properties of the solution may be much more complicated and the well-established properties for constant coefficient case would not be extended to variable-coefficient counterpart.The difficulties in finite element simulation lie in that(1)the coercivity of the Galerkin weak formulation mavnot hold,as pointed out by Wang and Yang[134,136];(2)solution of fractional diffusion equation contains singular terms which are typically parts of the solution and influenceheavily the regularity of the solution,as proved by Ervin etc.[37]All of these make the regularity of the solution more complicated.So the research on numerical simulation based on finite element framework develops slowly for fractional diffusion equation with variable coefficients,and the related research results are few.To overcome these difficulties,Hong Wang and Danping Yang developed a Petrov-Galerkin finite element procedure[137]with an optimal-order convergence,by utilizing the DPG(Discontinuous Petrov-Galerkin)framework of Demkowicz and Gopalakrish-nan,to approximate variable-coefficient diffusion equation with fractional flux and per-formed numerical experiments for sufficiently smooth solutions.This Petrov-Galerkin procedure has to analyze that how to choose appropriate finite-dimensional trial space and test space to ensure the weak coercivity and the unique solvability and stability of the corresponding Petrov-Galerkin finite element method,and also needs to constructa K(x)-dependent and nonlocal test function space from its trial function space by a nonlocal transformation to ensure that LBB constraint is satisfied.This makes its com?putation more complicated and increases the complexity of the approximation scheme.Therefore,we do think that it is a new challenge to develop a Galerkin variational formulation for determining and computing the singular term efficiently and inducing a finite element procedure with high convergence accuracy.The main work of this thesis is listed as follows.In chapter 2,we considered a one-side constant-coefficient diffusion equation with fractional fluxThe appearance of the weak singular term in the solution,which may result in weak singularity at x = O,leads to lower regularity of the solution,H1-?-?(?),whatever the regularity of the right term is,so to the lower order convergence rate,dependence of the regularity of the solution,and to the unsatisfactory numerical result for the numerical solution.The aims of this part are to separate the singular term from the equation by equation decomposition technique and to construct a full-regular new fraction diffusion equation.Based on this,we split the full-regular diffusion equation into a hybrid system by introducing two intermediate variables u =-K0Ix?q and q = Dp,established the saddle point frame,constructed an extended mixed variational formulation independence of singular terms,and proposed the corresponding extended mixed element discrete scheme.Equation decomposition technique eliminated successfully the influence of low regularity of singular term on convergence accuracy,and extended mixed theoretical framework enables us to approximate the unknown function p and the intermediate variables q,u.The main works are presented as follows:(1)Split the fractional diffusion equa.tion into two equations:one is a f-dependent fractional diffusion equation that permits a fully regular solution,and the other is an analytic-solved fractional equation from which the x1-?-type analytic solution is easily solved.By doing so,the solution p is expressed as the sum of the f-dependent solution and the x1-?-type solution:(2)Devoted to discretizing the f-dependent fractional equation by expanded mixed finite element procedure and deriving the related error estimates.Obviously the error estimates for the original fractional diffusion equation obey those for the f-dependentfractional equation since the x1-?-type solution is analytically solved.Numerical ex?periments are performed to confirm our theoretical findings.In chapter 3,we considered a one-side variable-coefficient fractional diffusion equa?tion with fractional divergenceBecause of the appearance of the variable-coefficient K(x),the well-established coercivity of the Galerkin weak formulation for constant coefficient case would not be extended to variable-coefficient counterpart.We introduced u =-KDp as an intermediate variable to isolate K(x)from the nonlocal operator and to combine variable coefficient with integer derivative,and constructed a mixed system formulated by a one-order equation with variable-coefficient and a low order equation with constant coefficient.This allows us to deal with the two equations by Hl-Galerkin mixed methods under the Galerkin frame.Besides,the above equation often permits weak singular term,dependence on the variable-coefficient K(x),to be a part of the solution,which leads to the lower regularity of the exact solution and the lower convergence rates,dependence on the regularity of the exact solution.To overcome this difficulty,we made a direct-sum decomposition for general fractional derivative spaces JL1?(?)(admissible space of the solution)into a fractional Sobolev space HL1-?)and a singular space spanned by x? by applying space decomposition technique,and designed a x-?-independent,and thus x1-?-in-dependent,mixed type variational formulation over the commonly used Sobolev spaces based on least-squared technique.The direct-sum decomposition allows us to split the solution as the sum of a regular part in HL1-?)which can be approximated by a delicately selected least-squared mixed finite element(LSMFE)and a x-?-like term which can be computed exactly.This enables us to establish an easily-computed and optimal-order-convergent Galerkin finite element method.The main works are presented as follows:(1)Made a direct-sum decomposition for general fractional derivative spaces JLs(?)by delicately analyzing the relationship between fractional spaces JlS(?)and Hs(?),and the properties of fractional operators 0Ixs and 0Dxs;(2)Constructed a mixed system formulated by a one-order equation with variable-coefficient and a low order equation with constant coefficient,by introducing u =-KDp as an intermediate variable.Established a x-?-independent mixed type variational for-mulation from least-squared technique and direct-sum decomposition,proved the exis-tence and uniqueness of the variational formulation,showed the equivalence between the variational formulation and the fractional diffusion equation and discussed the regularity of the solution to this equation with a general right hand side function f;(3)Developed a least-squared mixed finite element method,proved the optimal-order error estimates and conducted numerical experiments to verify the performance of the proposed LSMFE;(4)Constructed a spectral Galerkin algorithm based on the orthogonality of Ja-cobi polynomials to reduce the computation cost and storage of the linear equations corresponding to the least-squared mixed finite element formulation.In chapter 4,we considered the two-side variable-coefficient fractional diffusion e-quation with fractional divergenceThe well-established coercivity of the Galerkin weak formulation for constant coef?ficient case is lost for the variable-coefficient counterpart due to the appearance of the variable-coefficient K(x);the lower regularity of the exact solution leads to the lower convergence rates by the appearance of the weak singular terms ?(x),x?(x)in the exact solution,dependence of the variable-coefficient K(x).To remove the effect of variable-coefficient to the coercivity of the Galerkin weak formulation,we introduced u =—KDp as an intermediate variable to isolate K(x)from the nonlocal operator and to combine variable coefficient with integer derivative,and constructed a mixed system formulated by a one-order equation with variable-coefficient and a low order equation with constant coefficient.This allows us to deal with the two equations by H1-Galerkin mixed meth-ods under the Galerkin frame.To overcome the deficiency caused by two weak singular terms,we select the admissible space of fractional flux tx as a direct-sum of a fractional Sobolev space H01-?(?)and a singular space spanned by singular terms ?(x),x?(x),by applying the space decomposition technique,and designed a ??(x),x?(x)-independent mixed type variational formulation over the commonly used Sobolev spaces based on least-squared technique.This ensures the coercivity of the variational formulation and avoids the influence on convergence accuracy of the least-squared mixed element discrete formulation induced by the variational formulation caused by the appearance of singular terms in the variational formulation.The main works are presented as follows:(1)Proved the equivalence between the two-side fractional derivative space J?1-?(?)and the Sobolev space H01-?(?)(2)Discussed the kernel space,positive-definiteness and invertibility of the two-side fractional differential operator D?1-?;(3)Developed a least-squared mixed formulation,by introducing the intermediate variable,for the fractional diffusion equa.tion,in which the admissible space of the so-lution u is selected as a direct sum of H01-?(?)and a known singular-terms-panned space,so as to express the solution[u,p]by its regular part and two singular parts,and demonstrated the solvability and regularity of the solution;(4)Proposed a least-squared mixed finite element method,demonstrated its solv-ability and optimal-order convergence and conducted numerical experiments to verify the performance of the proposed methods.