A Mixed-type Galerkin Variational Formulation And Fast Algorithms For Variable-coefficient Fractional Diffusion Equations

In this paper, we consider the following fractional-order diffusion equation of order 2-? with 0 < ?< 1,-(?0D1x-? + (1 - ?)xD11-?)(K(x)Dc) = f,x ??= (0, 1),c(0) = c(1) = 0.Here the unknown c stands for the concentration of a solute, K(x) is the diffusivity coefficient with 0 < Kmin ? K(x) ? Kmax < +? , x ? (0, 1),f ? L2(?) is the source or sink term, and 0 < ? < 1 indicates the relative weight of forward versus backward transition probability. We let D be the first-order differ-ential operator,0D1-?x and xD1-?1 refer to respectively the left and right Riemann-Liouville fractional derivative operators of order 1 - ?.By introducing an intermediate variable u = -K(x)Dc, we propose a mixed-type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over H10(?)×H1-?/2(?) for variable-coefficient fractional-order diffusion equation with two-sided fractional derivatives. Under the assumption that· the solution has certain regularity, we establish the equivalence between (0.0.1) and the variational formulation. Based on the formulation we develop a mixed-type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable.Due to the nonlocal property of fractional derivative or integral operators,numerical methods of fractional diffusion equations generate full or dense matrices.Traditional methods to solve these numerical methods requires computational cost of O(N3) and O(N2) of memory, where N is the number of spatial grid points.To solve the mixed-type finite element formulation, we find out that the coefficient matrix is composed of four block matrices. They are a zero matrix, two tri-diagonal sparse matrices and a Toeplitz - like matrix. By noting that the computational work is O(N log N) and the memory requirement is O(N) when the fast Fourier transform is used to solve the matrix-vector multiplication with a Toeplitz coefficient matrix, we apply the fast Fourier transform and the conjugate gradient method to solve the algebraic equation, and then obtain the fast conjugate gradient algorithm with the O(N log N) computational work and the O(N) memory requirement per iteration. For reducing the number of the iterations, we apply the preconditioned technique to fast method. The use of this technique significantly reduces the number of iterations and ensures accuracy of convergence. The total computational work of this preconditioned method is still O(N log N). Numerical experiments show the good property of the preconditioned fast conjugate gradient method.