An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements

The numerical solution for a class of sub-diffusion equations involving a parameter in the range −1 < α < 0 is studied. For the time discretization, we use an implicit finite-difference Crank–Nicolson method and show that the error is of order k2+α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h2 max(1, log k−1), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.