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On nonnegativity preservation in finite element methods for subdiffusion equations


Authors: Bangti Jin, Raytcho Lazarov, Vidar Thomée and Zhi Zhou
Journal: Math. Comp. 86 (2017), 2239-2260
MSC (2010): Primary 65M12, 65M60
DOI: https://doi.org/10.1090/mcom/3167
Published electronically: December 21, 2016
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Abstract: We consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations, for which the maximum-principle holds and which, in particular, preserve nonnegativity. Hence the solution is nonnegative for nonnegative initial data. Following earlier work on the heat equation, our purpose is to study whether this property is inherited by certain spatially semidiscrete and fully discrete piecewise linear finite element methods, including the standard Galerkin method, the lumped mass method and the finite volume element method. It is shown that, as for the heat equation, when the mass matrix is nondiagonal, nonnegativity is not preserved for small time or time-step, but may reappear after a positivity threshold. For the lumped mass method nonnegativity is preserved if and only if the triangulation in the finite element space is of Delaunay type. Numerical experiments illustrate and complement the theoretical results.


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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

Raytcho Lazarov
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: lazarov@math.tamu.edu

Vidar Thomée
Affiliation: Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Göteborg, Sweden
Email: thomee@chalmers.se

Zhi Zhou
Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, 500 W. 120th Street, New York, New York 10027
Email: zhizhou0125@gmail.com

DOI: https://doi.org/10.1090/mcom/3167
Keywords: Subdiffusion, finite element method, nonnegativity preservation, Caputo fractional derivative
Received by editor(s): October 10, 2015
Received by editor(s) in revised form: March 19, 2016
Published electronically: December 21, 2016
Additional Notes: The work of the first author was partially supported by UK Engineering and Physical Sciences Research Council grant EP/M025160/1.
Article copyright: © Copyright 2016 American Mathematical Society