A fast solution technique for finite element discretization of the space-time fractional diffusion equation

In this paper, we study fast Galerkin finite element methods to solve a space-time fractional diffusion equation. We develop an optimal piecewise-linear and piecewise-quadratic finite element methods for solving this problem and give optimal error estimates. Furthermore, we develop piecewise-constant discontinuous finite element method for discontinuous problem of this model. Importantly, a fast solution technique to accelerate non-square Toeplitz matrix-vector multiplications which arise from both continuous and discontinuous Galerkin finite element discretization respectively is considered. This fast solution technique is based on fast Fourier transform and depends on the special structure of coefficient matrices and it helps to reduce the computational work from O(N3) required by the traditional methods to O(Nlog2⁡N), where N is the size (number of spatial grid points) of the coefficient matrices for every time step. Moreover, the applicability and accuracy of the method are demonstrated by numerical experiments to support our theoretical analysis.