L1/finite element methods for time-fractional Keller-Segel equations with weak singularity solution

In this paper, we consider the L1/ finite element methods for solving time-fractional Keller-Segel equations with a singular solution. For time direction, the L1 scheme under the nonuniform mesh is considered to approximate the Caputo derivative to handle the singularity of the solution. To further reduce computational storage requirements, we consider a fast L1 scheme based on the sum-of-exponentials skill for calculating fractional derivatives. Then, we derive a fully implicit discrete scheme by combining the finite element method in the spatial direction. Subsequently, unconditional stability and $\alpha$-robust error estimates of the fully discrete scheme are derived. Finally, numerical examples are presented to verify our theory.