A Fast L2-1σ Finite Element Method for Time Fractional Keller–Segel Equations with Weakly Singular Solutions

We propose a fast L2-1σ finite element method for solving the time fractional Keller–Segel equations with a Caputo fractional derivative of α∈(0,1). Firstly, the fast L2-1σ scheme on the graded mesh is used to discretize the time fractional derivative. This approach relies on the sum of exponentials (SOE) skill to speed up the convolution kernel. Thus, we overcome the computational cost caused by the nonlocality of fractional derivatives. Then, by combining finite element discretization in spatial direction, a fully implicit numerical scheme is derived. Subsequently, we establish the stability and an α-robust error analysis of the fully discrete scheme. Finally, we present some numerical examples to demonstrate the correctness of our theoretical results.