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 < 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 super(2+) alpha where k denotes the maximum...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2011-04, Vol.31 (2), p.719-739
Main Author: Mustapha, K.
Format: Article
Language:English
Subjects:
Discretization
Errors
Exact solutions
Finite difference method
Finite element method
Mathematical analysis
Mathematical models
Nonuniform
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Summary: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 super(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 super(2) max(1, log k super(-1)), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drp057