A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions

We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O ( M N log 3 ⁡ N ) computational complexity and O ( M N log 2 ⁡ N ) memory with M and N being the numbers of degrees of freedom in space and time, respectively. Furthermore, a preconditioner is introduced to reduce the number of iterations caused by the bad condition number of the coefficient matrix. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.