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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On nonnegativity preservation in finite element methods for subdiffusion equations
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by Bangti Jin, Raytcho Lazarov, Vidar Thomée and Zhi Zhou PDF
Math. Comp. 86 (2017), 2239-2260 Request permission

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
  • MR Author ID: 741824
  • 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
  • MR Author ID: 111240
  • Email: lazarov@math.tamu.edu
  • Vidar Thomée
  • Affiliation: Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Göteborg, Sweden
  • MR Author ID: 172250
  • 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
  • MR Author ID: 1011798
  • Email: zhizhou0125@gmail.com
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 2239-2260
  • MSC (2010): Primary 65M12, 65M60
  • DOI: https://doi.org/10.1090/mcom/3167
  • MathSciNet review: 3647957