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

作者：Mustapha Kassem^*

来源：IMA Journal of Numerical Analysis, 2011, 31(2): 719-739.

DOI：10.1093/imanum/drp057

• sub-diffusion (fractional diffusion)
• nonuniform time steps
• finite-difference method
• finite-element method

The numerical solution for a class of sub-diffusion equations involving a parameter in the range - 1 < alpha < 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+alpha), 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 h(2) max(1, log k(-1)), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.

• 出版日期2011-4
• 单位中国石油大学（北京）