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On nonnegativity preservation in finite element methods for subdiffusion equations

Jin, B; Lazarov, R; Thomee, V; Zhou, Z; (2017) On nonnegativity preservation in finite element methods for subdiffusion equations. Mathematics of Computation , 86 (307) pp. 2239-2260. 10.1090/mcom/3167. Green open access

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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.

Type: Article
Title: On nonnegativity preservation in finite element methods for subdiffusion equations
Location: USA
Open access status: An open access version is available from UCL Discovery
DOI: 10.1090/mcom/3167
Publisher version: https://doi.org/10.1090/mcom/3167
Language: English
Additional information: © Copyright 2016 American Mathematical Society. This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: subdiffusion, finite element method, nonnegativity preservation, Caputo derivative
UCL classification: 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: http://discovery.ucl.ac.uk/id/eprint/1477538
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