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Subdiffusion with a time-dependent coefficient: Analysis and numerical solution


Authors: Bangti Jin, Buyang Li and Zhi Zhou
Journal: Math. Comp. 88 (2019), 2157-2186
MSC (2010): Primary 65M30, 65M15, 65M12
DOI: https://doi.org/10.1090/mcom/3413
Published electronically: February 6, 2019
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Abstract: In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.


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Additional Information

Bangti Jin
Affiliation: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom
Email: bangti.jin@gmail.com, b.jin@ucl.ac.uk

Buyang Li
Affiliation: Department of Applied Mathematics, The Polytechnic University of Hong Kong, Kowloon, Hong Kong
Email: bygli@polyu.edu.hk

Zhi Zhou
Affiliation: Department of Applied Mathematics, The Polytechnic University of Hong Kong, Kowloon, Hong Kong
Email: zhizhou@polyu.edu.hk

DOI: https://doi.org/10.1090/mcom/3413
Keywords: Subdiffusion, time-dependent coefficient, Galerkin finite element method, convolution quadrature, perturbation argument, error estimate
Received by editor(s): April 17, 2018
Received by editor(s) in revised form: September 10, 2018
Published electronically: February 6, 2019
Additional Notes: The research of the second author was partially supported by a Hong Kong RGC grant (Project No. 15300817).
The research of the third author was supported by a start-up grant from the Hong Kong Polytechnic University and Hong Kong RGC grant No. 25300818.
Article copyright: © Copyright 2019 American Mathematical Society