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

Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount...

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Published in:Computer methods in applied mechanics and engineering Vol. 346; pp. 332 - 358
Main Authors: Jin, Bangti, Lazarov, Raytcho, Zhou, Zhi
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.04.2019
Elsevier BV
Subjects:
Computer simulation
Convolution
Error estimates
Evolution
Finite element method
Formulations
Galerkin method
Inverse problems
Mathematical analysis
Mathematical models
Nonsmooth solution
Numerical analysis
Numerical methods
Optimal control
Space–time formulation
Time-fractional evolution
Time-stepping
Finite element method
Time-stepping
Nonsmooth solution
Space–time formulation
Error estimates
Time-fractional evolution
ISSN:0045-7825, 1879-2138
Online Access:Get full text
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Summary:Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) 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.
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ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2018.12.011