Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well. The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near $t=0$. In this paper, we propose a modification. We first split the time interval $[0,T]$ into $[0,T_0]$ and $[T_0,T]$, where $T_0 (0<T_0<T)$ is reasonably small. Then, the graded L1 scheme is applied in $[0,T_0]$, while the uniform one is used in $[T_0,T]$. Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.

Dr. Yong-Liang Zhao is an assistant professor at the Sichuan Normal University. He obtained his Ph.D degree from University of Electronic Science and Technology of China. From Sep. 2019 to Sep. 2020, he was funded by the China Scholarship Council to study at the University of Innsbruck. His main research interests are fast (PinT) numerical solutions of (F)PDEs, Dynamical low-rank approximations and Exponential integrators. He has published over 20 papers in journals such as J. Comput. Phys., J. Sci. Comput., BIT, Numer. Algorithms and Numer. Methods PDE.