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

作者：Liu, Zhengguang; Cheng, Aijie^*; Li, Xiaoli; Wang, Hong

来源：Applied Numerical Mathematics, 2017, 119: 146-163.

DOI：10.1016/j.apnum.2017.04.003

• Fast Galerkin methods
• Space-time fractional diffusion equation
• Toeplitz matrix
• Fast Fourier transform

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(N-3) required by the traditional methods to O(N log(2)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.

• 出版日期2017-9
• 单位山东大学