Expanded Mixed Finite Element Formulation For Time-space Fractional Diffusion Equations

In the first part of this thesis,we consider the following fractional-order conservative diffusion equation of order 2-? with 0<?<1 Here,? =(0,1),0<?<1,0 ??<1,K is the difiusivity coefficient,f ? L2(?)is the source or sink term;(?)and D= d/dx are the first derivative operators for time and space.0Ix? and xI1? refer to the left and right Riem1nn-Liouville fractional integral operators of order ? defined below by(2.2.1).respectively.A large number of experiments have shown that the above fractional diffu-sion equation can describe the anomalous diffusion or non-Fick phenomenon,such as turbulence,chaotic dynamics,viscoelastic mechanics,underground contaminant seepage,more accurately than the second-order diffusion equation.It has a dis-tinct physical background and a wide range of applications.Since it is difficult to use Fourier transform,Laplace transform and other methods to obtain the analyt-ical solution of general fractional diffusion problem,one usually uses finite element method and finite difference method to find the numerical solution.To comply with engineering needs,an ideal numerical method should not only pay attention to the unknown function and its flux,but also maintain conservative to reflect the mathematical physical characteristics of the material diffusion trans-port process.The idea is to establish a proper saddle point variational principle by introducing the fractional order flux p =-K(?0Ix?+(1-?)xI1?)Du as an interme-diate variable to construct a standard mixed finite element method.But because of the operators 0Ix? Du,xI1?Du and D are not symmetric,we can't construct an appropriate bilinear form b(*,·)to satisfy the in f-sup condition.By introducing q = Du as the second intermediate variable,Chen Huanzhen and Wang Hong[10]has established saddle point principle for the space fractional diffusion equation.Based on this,an expanded mixed finite element method that can approximate the unknown function and the fractional order flux with high accuracy and maintain the cell conservation is,constructed.In this thesis,we borrow the idea of[10]and introduce the fractional-order flux and the derivative of the unknown function as intermediate variables to establish the corresponding saddle-point variational formulation for the time-dependent fraction-al diffusion equation.We construct an expanded mixed finite element scheme that satisfies local conservation and can accurately approximate the unknown function:the derivative of the unknown function and the fractional order flux.By using the backward Euler method to discretize the time derivative,we establish a fully discrete expanded mixed finite element method.Further,we use the invertibility of the coef-ficient matrix of the derived mixed equations to prove the existence and uniqueness of the full-discrete scheme.Using the discrete Growill inequality,we prove the un-conditional stability of the discrete scheme.In numerical analysis,we use the error estimate of the mixed element projection operator to get the convergence order of the fully discrete expanded mixed finite element method is O(?+hmin{k+1,s-1+?/2}),where ?,h and k represent the time step,space step and finite element space expo-nent,respectively.Numerical experiments show that the expanded mixed element method proposed in this thesis can be effectively simulated by the above anomalous diffusion transport model.However,a large number of anomalous diffusion or non-Fick transport processes are characterized by the phenomenon of super-diffusion.That is,the derivatives of time and space are fractional.Therefore,in the second part of this thesis,we mainly introduce the time-space fractional diffusion.Here(?)?u(x,t)represents the Caputo fractional derivative of order ?,0<?<1.By introducing the flux of the unknown function p =-K(?0Ix?+(1-?)xI1?)Du and the derivative q = Du as intermediate variables,we establish the corresponding saddle point variation formulation and the expanded mixed finite element scheme.On the basis of the formulation we constructed the L1 fully discrete expanded mixed finite element scheme.In the numerical analysisi we adopt the idea of mathemat-ical induction to replace the discrete Gronwall inequality.Through the complex analysis and argumentation,we establish the convergence theory of the fully dis-crete expanded mixed element method and obtain the convergence order of u is O(?2-?+hmin{k+1,s-1+?/2}),the convergence order of the function derivative and the fractional numerical flux p is O(?2-3?/2+h min{k-1,s-1+?/2).Numerical experiments in the thesis show that the proposed L1 fully discrete expanded mixed finite element scheme has an ideal numerical approximation effect.