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 k^2+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.