An $\alpha$-robust second-order accurate scheme for a subdiffusion equation

We investigate a second-order accurate time-stepping scheme for solving atime-fractional diffusion equation with a Caputo derivative of order~$\alpha\in (0,1)$. The basic idea of our scheme is based on local integration followedby linear interpolation. It reduces to the standard Crank--Nicolson scheme inthe classical diffusion case, that is, as $\alpha\to 1$. Using a novelapproach, we show that the proposed scheme is $\alpha$-robust and second-orderaccurate in the $L^2(L^2)$-norm, assuming a suitable time-graded mesh. Forcompleteness, we use the Galerkin finite element method for the spatialdiscretization and discuss the error analysis under reasonable regularityassumptions on the given data. Some numerical results are presented at the end.