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Numerical analysis of nonlinear subdiffusion equations

Jin, B; Li, B; Zhou, Z; (2018) Numerical analysis of nonlinear subdiffusion equations. SIAM Journal on Numerical Analysis , 56 (1) pp. 1-23. 10.1137/16M1089320. Green open access

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Abstract

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α ∈ (0, 1) in time. It relies on three technical tools: a fractional version of the discrete Grönwall type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Grönwall inequality and verify it for the L1 scheme and convolution quadrature generated by backward difference formulas. Further, we provide a complete solution theory, e.g., existence, uniqueness, and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise L2 (Ω) norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order O(h 2 ) (up to a logarithmic factor) and O(τ α), respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments.

Type: Article
Title: Numerical analysis of nonlinear subdiffusion equations
Open access status: An open access version is available from UCL Discovery
DOI: 10.1137/16M1089320
Publisher version: http://dx.doi.org/10.1137/16M1089320
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: nonlinear fractional diffusion equation, discrete fractional Grönwall inequality, L1 scheme, convolution quadrature, error estimate
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/1561160
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