Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] P. Chatzipantelidis, Z. Horváth, and V. Thomée, On preservation of positivity in some finite element methods for the heat equation, Comput. Methods Appl. Math. 15 (2015), no. 4, 417-437. MR 3403443
  • [2] Panagiotis Chatzipantelidis, Raytcho Lazarov, and Vidar Thomée, Some error estimates for the finite volume element method for a parabolic problem, Comput. Methods Appl. Math. 13 (2013), no. 3, 251-279. MR 3094617, https://doi.org/10.1515/cmam-2012-0006
  • [3] A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002), 046129.
  • [4] So-Hsiang Chou and Q. Li, Error estimates in $ L^2,\ H^1$ and $ L^\infty $ in covolume methods for elliptic and parabolic problems: a unified approach, Math. Comp. 69 (2000), no. 229, 103-120. MR 1680859, https://doi.org/10.1090/S0025-5718-99-01192-8
  • [5] Andrei Drăgănescu, Todd F. Dupont, and L. Ridgway Scott, Failure of the discrete maximum principle for an elliptic finite element problem, Math. Comp. 74 (2005), no. 249, 1-23 (electronic). MR 2085400, https://doi.org/10.1090/S0025-5718-04-01651-5
  • [6] William Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
  • [7] H. Fujii, Some remarks on finite element analysis of time-dependent field problems, In. Theory and Practice in Finite Element Structural Analysis (Y. Yamada, R. H. Gallagher & N. K. Kyokai eds). Tokyo, Japan: University of Tokyo Press, 1973, pp. 91-106.
  • [8] Bangti Jin, Raytcho Lazarov, Yikan Liu, and Zhi Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys. 281 (2015), 825-843. MR 3281997, https://doi.org/10.1016/j.jcp.2014.10.051
  • [9] Bangti Jin, Raytcho Lazarov, Dongwoo Sheen, and Zhi Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal. 19 (2016), no. 1, 69-93. MR 3475410, https://doi.org/10.1515/fca-2016-0005
  • [10] Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), no. 1, 445-466. MR 3033018, https://doi.org/10.1137/120873984
  • [11] Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput. 38 (2016), no. 1, A146-A170. MR 3449907, https://doi.org/10.1137/140979563
  • [12] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
  • [13] Anatoly N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl. 340 (2008), no. 1, 252-281. MR 2376152, https://doi.org/10.1016/j.jmaa.2007.08.024
  • [14] Zhiyuan Li, Yikan Liu, and Masahiro Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput. 257 (2015), 381-397. MR 3320678, https://doi.org/10.1016/j.amc.2014.11.073
  • [15] Zhiyuan Li, Yuri Luchko, and Masahiro Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fract. Calc. Appl. Anal. 17 (2014), no. 4, 1114-1136. MR 3254683, https://doi.org/10.2478/s13540-014-0217-x
  • [16] Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533-1552. MR 2349193, https://doi.org/10.1016/j.jcp.2007.02.001
  • [17] C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129-145. MR 923707, https://doi.org/10.1007/BF01398686
  • [18] Yury Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal. 12 (2009), no. 4, 409-422. MR 2598188
  • [19] Yury Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl. 351 (2009), no. 1, 218-223. MR 2472935, https://doi.org/10.1016/j.jmaa.2008.10.018
  • [20] Yury Luchko, Initial-boundary problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl. 374 (2011), no. 2, 538-548. MR 2729240, https://doi.org/10.1016/j.jmaa.2010.08.048
  • [21] K. Mustapha, B. Abdallah, and K. M. Furati, A discontinuous Petrov-Galerkin method for time-fractional diffusion equations, SIAM J. Numer. Anal. 52 (2014), no. 5, 2512-2529. MR 3270028, https://doi.org/10.1137/140952107
  • [22] Per-Olof Persson and Gilbert Strang, A simple mesh generator in Matlab, SIAM Rev. 46 (2004), no. 2, 329-345 (electronic). MR 2114458, https://doi.org/10.1137/S0036144503429121
  • [23] Harry Pollard, The completely monotonic character of the Mittag-Leffler function $ E_a(-x)$, Bull. Amer. Math. Soc. 54 (1948), 1115-1116. MR 0027375
  • [24] Kenichi Sakamoto and Masahiro Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426-447. MR 2805524, https://doi.org/10.1016/j.jmaa.2011.04.058
  • [25] A. H. Schatz, V. Thomée, and L. B. Wahlbin, On positivity and maximum-norm contractivity in time stepping methods for parabolic equations, Comput. Methods Appl. Math. 10 (2010), no. 4, 421-443. MR 2770304, https://doi.org/10.2478/cmam-2010-0025
  • [26] J. R. Shewchuk, Triangle: Engineering a 2d quality mesh generator and delaunay triangulator, Applied Computational Geometry: Towards Geometric Engineering (Ming C. Lin and Dinesh Manocha, eds.), Springer, 1996, pp. 203-222. MR 1445297
  • [27] V. Thomée, On positivity preservation in some finite element methods for the heat equation, Numerical methods and applications, Lecture Notes in Comput. Sci., vol. 8962, Springer, Cham, 2015, pp. 13-24. MR 3334845, https://doi.org/10.1007/978-3-319-15585-2_2
  • [28] Vidar Thomée and Lars B. Wahlbin, On the existence of maximum principles in parabolic finite element equations, Math. Comp. 77 (2008), no. 261, 11-19 (electronic). MR 2353941, https://doi.org/10.1090/S0025-5718-07-02021-2
  • [29] J. A. C. Weideman and L. N. Trefethen, Parabolic and hyperbolic contours for computing the Bromwich integral, Math. Comp. 76 (2007), no. 259, 1341-1356 (electronic). MR 2299777, https://doi.org/10.1090/S0025-5718-07-01945-X

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M12, 65M60

Retrieve articles in all journals with MSC (2010): 65M12, 65M60


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

American Mathematical Society