An Expanded Mixed Finite Element Method For The Conservative Fractional Diffusion Equation

In this dissertation, we focus on the numerical simulation for the following conservative fractional-order diffusion equation with two-sided fractional derivatives, Here K is the diffusivity coefficient and f ? L2(?) is the source or sink term; D=d/dx is the first-order derivative operator,0Ix? and xI1? refer to the left and right Riemann-Liouville fractional integral operators of order (3 defined below by (2.2.1) and (2.2.3). 0???1.?=0 or ?=1 corresponds to the one-sided equation governed by left or right Riemann-Liouville fractional differential operators respectively.This equation describes the well-known anomalous or non-Fickian diffusion processes, in which the diffusion depends on its global behaviors. From the point of view of numerical issue, an ideal numerical procedure should recognize both the unknown function and its flux to comply with engineering needs, and obey the conservation law to reflect the physical character of the diffusion model. Compared to second-order diffusion, the coefficient matrix generated by the discrete procedure of this equation is non-sparse and so its computation is costly because the fractional operators possess non-local characteristic. It is a challenge to develop fast numerical algorithm to approximate the fractional diffusion equations.For this, based on the saddle-point theory and the negative-order fractional derivative spaces, we introduce the fractional-order flux p=-K(?0Ix?+(1-?)xI1?)q and its derivative q=Du of the unknown as auxiliary variables, and establish a mixed variational principle and prove its equivalence H1(?)×H-?/2)(?)×L2 (?) to the fractional diffusion equation. Upon the variational principle we design a locally-conservative expanded mixed finite element procedure to approximate the unknown function, its derivative and the fractional flux directly. Our mathematical analysis shows that the discrete procedure conserve the mass element by element which is important for computation in engineering; and the unique discrete solution possesses the optimal-order convergence rate in L2 or H-?/2-norm, in some sense. In numerical analysis, we abandon the dual argument being used in [16] with an unproven regularity assumption for the adjoint problem of the fractional diffusion equation, and instead, apply the merits of the projection of the exact solution. By doing so, we establish the optimal-order convergence theory without the unproven regularity assumption. A list of numerical experiments are conducted to confirm the validity of the expanded mixed finite element procedure.Due to the nonlocal property of fractional differential operators, their numeri-cal methods often generate coefficient matrices with complicated structures. Tradi-tional Gaussian elimination requires computational work of O(N3) and memory of O(N2). The huge amount of computational work and memory makes the simulation extremely expensive when N is large enough. Therefore, it is necessary to develop a fast algorithm to numerically solve them. Luckily, we find out that the stiff ma-trix of discrete procedure can be decomposed into four zero matrices, four sparse matrices and a Toeplitz matrix, and that using fast Fourier transform to solve the matrix-vector multiplication with a Toeplitz coefficient matrix will result in an ideal computational work O(N log N). In view of this fact, we combine fast Fourier trans-form with conjugate gradient method and design a fast conjugate gradient method (FCG) for the expanded mixed finite element algorithm with O(N log N) compu-tational work and O(N) memory requirement per iteration. However, the amount of iterations in this FCG is heavily depend on the amount of unknowns due to the ill-conditioned matrix, which makes the computational cost of FCG is unsatisfac-tory. To improve, we combine the preconditioned technique with FCG to construct a preconditioned fast conjugate gradient method (PFCG) for the expanded mixed finite element method. Numerical experiments conducted in this dissertation show that the amount of iterations of PFCG doesn't depend on the counterpart of un-knowns, and the constructed PFCG possesses ideal computational work O(N log N) and memory of O(N).