Mixed-type Finite Element Method For Space-time Fractional Diffusion Equation With Variable Coefficients

We consider the following space-time fractional diffusion equations with variable-coefficient where D is the first-order differential operator,0Dx1-? and xD11-? refer to,respec-tively,the left and right Riemann-Liouville fractional derivative operators of or-der 1-?(0<?<1).0CDt?u is Caputo fractional derivative operator of order?(0<?<1).when ? = 1,0CDt?u=ut is the first-order differential of time;K(x)is the diffusivity coefficient with satisfying f?L2(?)is the source or sink term,and 0 ? ? ? 1 indicates the relative weight of forward versus backward transition probability.We can not directly use Galerkin finite element to discrete the variable-coefficient fractional equation(0.0.1)since the corresponding bilinear form may not be coer-cive as K(x)depends on x.Hence,we are forced to turn to mixed Galerkin finite element approach.In this thesis,by introducing the flux function p =-K(x)Du as an intermediate variable,we split(0.0.1)into a coupled system of an integer order equation with variable-coefficient K(x)and a fractional differential equation with constant coefficient,and derive the mixed-type weak formulation.We apply mixed type finite element or mixed-type discontinuous finite element to discrete space frac-tional derivatives,and use the backward Euler scheme for ? = 1 or L1-scheme for 0<?<1 to discrete the temporal derivative,so as to formulate fully discrete mixed-type finite element schemes.Further,we use delicately the algebraic prop-erties of antisymmetric matrix and blocked matrix to prove the invertibility of the coefficient matrix resulted from the fully discrete mixed-type finite element scheme,then the existence and uniqueness of the fully discrete mixed-type finite element solution is proved.In the process of solving finite element equations,we find that,due to the mixed formulation and the non-locality of the fractional derivatives,its coefficient matrix is nonsymmetric and non-sparse,which will generate the computational cost and storage up to O(N3)and O(N2)when the traditional Gauss elimination is used.To overcome this deficiency,we introduce the fast Fourier transform(FFT)on each Toeplitz block matrix to reduce the computational cost and storage for each block matrix to O(N log N)and O(N).Thus,the storage requirement of full coefficient matrices is significantly reduced from O(N2)to O(N),computational cost reduces from O(N3)to O(N2).Notice that the computation efficiency also depends on the number of per iter-ation which is influenced heavily by bad condition number of the coefficient matrix since it is nonsymmetric.By replacing the traditional preconditioner for Toeplitz block matrix by a new circulant preconditioner and embedding it into the stable bi-conjugate gradient method(BiCG-STAB),we thus propose the preconditioned fast stable bi-conjugate gradient algorithm(PFBiCG)to approximate numerically the space-time fractional diffusion equation with variable coefficients.The numeri-cal experiments show that the PFBiCG keeps the number of per iteration constant and the computation efficiency is improved obviously,compared with the traditional Guass elimination and the stable bi-conjugate gradient method(BiCG-STAB).