A fast algorithm for time-fractional diffusion equation with space-time-dependent variable order

We investigate a fast algorithm for the time-fractional diffusion equation with a space-time-dependent variable order, which models, e.g., the subdiffusion with varying memory effects. In addition to the traditional L1 discretization of the time-fractional derivative, we perform a further approximation for the L1 coefficients, analyze the structures of the resulting all-at-once system, and apply the divide and conquer method to obtain a fast numerical algorithm. Due to the spatial dependence of the variable order and the further approximation to the L1 coefficients, the temporal discretization coefficients are coupled with the inner product of the finite element method and lack the monotonicity, which are rarely encountered in previous works and thus motivate novel analysis methods and computational techniques. Compared with the standard time-stepping methods with L1 discretization, the proposed algorithm reduces the complexity of solving the all-at-once system from O(MN^2) to O(MNln ^3 N) , where M stands for the spatial degree of freedom and N refers to the number of time steps. Numerical experiments are provided to substantiate the theoretical findings.