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Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

Jin, B; Lazarov, R; Zhou, Z; (2019) Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview. Computer Methods in Applied Mechanics and Engineering , 346 pp. 332-358. 10.1016/j.cma.2018.12.011. Green open access

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Abstract

Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following topics of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space–time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.

Type: Article
Title: Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.cma.2018.12.011
Publisher version: https://doi.org/10.1016/j.cma.2018.12.011
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Time-fractional evolution, Nonsmooth solution, Finite element method, Time-stepping, Error estimates, Space–time formulation
UCL classification: UCL
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/10063344
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