On nonnegativity preservation in finite element methods for subdiffusion equations. (English) Zbl 1364.65197
The authors consider the numerical approximation of three kinds 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. The aim of this work is to study whether this property, which holds for the heat equation, is maintained 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 proved 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. Some numerical experiments are presented to illustrate and complement the theoretical results.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
| 35R11 | Fractional partial differential equations |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 35B50 | Maximum principles in context of PDEs |
Keywords:
subdiffusion; finite element method; nonnegativity preservation; Caputo fractional derivative; fractional diffusion equations; maximum-principle; heat equation; semidiscrete; fully discrete; Galerkin method; lumped mass method; finite volume element method; numerical experimentsReferences:
| [1] | Chatz:2014 P. Chatzipantelidis, Z. Horv\'ath, and V. Thom\'ee, On preservation of positivity in some finite element methods for the heat equation, Comput. Methods Appl. Math. 15 (2015), no. 4, 417-437. · Zbl 1327.65172 |
| [2] | Chatzipantelidis, Panagiotis; Lazarov, Raytcho; Thom{\'e}e, Vidar, Some error estimates for the finite volume element method for a parabolic problem, Comput. Methods Appl. Math., 13, 3, 251-279 (2013) · Zbl 1285.65059 · doi:10.1515/cmam-2012-0006 |
| [3] | ChechkinGorenfloSokolov:2002 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] | Chou, So-Hsiang; Li, Q., Error estimates in \(L^2,\ H^1\) and \(L^\infty\) in covolume methods for elliptic and parabolic problems: a unified approach, Math. Comp., 69, 229, 103-120 (2000) · Zbl 0936.65127 · doi:10.1090/S0025-5718-99-01192-8 |
| [5] | Dr{\u{a}}g{\u{a}}nescu, Andrei; Dupont, Todd F.; Scott, L. Ridgway, Failure of the discrete maximum principle for an elliptic finite element problem, Math. Comp., 74, 249, 1-23 (electronic) (2005) · Zbl 1074.65129 · doi:10.1090/S0025-5718-04-01651-5 |
| [6] | Feller, William, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, xxiv+669 pp. (1971), John Wiley & Sons, Inc., New York-London-Sydney · Zbl 0219.60003 |
| [7] | Fujii:1973 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. · Zbl 0373.65047 |
| [8] | Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281, 825-843 (2015) · Zbl 1352.65350 · doi:10.1016/j.jcp.2014.10.051 |
| [9] | Jin, Bangti; Lazarov, Raytcho; Sheen, Dongwoo; Zhou, Zhi, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal., 19, 1, 69-93 (2016) · Zbl 1333.65111 · doi:10.1515/fca-2016-0005 |
| [10] | Jin, Bangti; Lazarov, Raytcho; Zhou, Zhi, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51, 1, 445-466 (2013) · Zbl 1268.65126 · doi:10.1137/120873984 |
| [11] | Jin, Bangti; Lazarov, Raytcho; Zhou, Zhi, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput., 38, 1, A146-A170 (2016) · Zbl 1381.65082 · doi:10.1137/140979563 |
| [12] | Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, xvi+523 pp. (2006), Elsevier Science B.V., Amsterdam · Zbl 1092.45003 |
| [13] | Kochubei, Anatoly N., Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340, 1, 252-281 (2008) · Zbl 1149.26014 · doi:10.1016/j.jmaa.2007.08.024 |
| [14] | Li, Zhiyuan; Liu, Yikan; Yamamoto, Masahiro, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput., 257, 381-397 (2015) · Zbl 1338.35471 · doi:10.1016/j.amc.2014.11.073 |
| [15] | Li, Zhiyuan; Luchko, Yuri; Yamamoto, Masahiro, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations, Fract. Calc. Appl. Anal., 17, 4, 1114-1136 (2014) · Zbl 1312.35184 · doi:10.2478/s13540-014-0217-x |
| [16] | Lin, Yumin; Xu, Chuanju, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [17] | Lubich, C., Convolution quadrature and discretized operational calculus. I, Numer. Math., 52, 2, 129-145 (1988) · Zbl 0637.65016 · doi:10.1007/BF01398686 |
| [18] | Luchko, Yury, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12, 4, 409-422 (2009) · Zbl 1198.26012 |
| [19] | Luchko, Yury, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351, 1, 218-223 (2009) · Zbl 1172.35341 · doi:10.1016/j.jmaa.2008.10.018 |
| [20] | Luchko, Yury, Initial-boundary problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374, 2, 538-548 (2011) · Zbl 1202.35339 · doi:10.1016/j.jmaa.2010.08.048 |
| [21] | Mustapha, K.; Abdallah, B.; Furati, K. M., A discontinuous Petrov-Galerkin method for time-fractional diffusion equations, SIAM J. Numer. Anal., 52, 5, 2512-2529 (2014) · Zbl 1323.65109 · doi:10.1137/140952107 |
| [22] | Persson, Per-Olof; Strang, Gilbert, A simple mesh generator in Matlab, SIAM Rev., 46, 2, 329-345 (electronic) (2004) · Zbl 1061.65134 · doi:10.1137/S0036144503429121 |
| [23] | Pollard, Harry, The completely monotonic character of the Mittag-Leffler function \(E_a(-x)\), Bull. Amer. Math. Soc., 54, 1115-1116 (1948) · Zbl 0033.35902 |
| [24] | Sakamoto, Kenichi; Yamamoto, Masahiro, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 1, 426-447 (2011) · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058 |
| [25] | Schatz, A. H.; Thom{\'e}e, V.; Wahlbin, L. B., On positivity and maximum-norm contractivity in time stepping methods for parabolic equations, Comput. Methods Appl. Math., 10, 4, 421-443 (2010) · Zbl 1283.65086 · doi:10.2478/cmam-2010-0025 |
| [26] | shewchuk1996triangle 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. |
| [27] | Thom{\'e}e, V., On positivity preservation in some finite element methods for the heat equation. Numerical methods and applications, Lecture Notes in Comput. Sci. 8962, 13-24 (2015), Springer, Cham · Zbl 1360.65243 · doi:10.1007/978-3-319-15585-2\_2 |
| [28] | Thom{\'e}e, Vidar; Wahlbin, Lars B., On the existence of maximum principles in parabolic finite element equations, Math. Comp., 77, 261, 11-19 (electronic) (2008) · Zbl 1128.65085 · doi:10.1090/S0025-5718-07-02021-2 |
| [29] | Weideman, J. A. C.; Trefethen, L. N., Parabolic and hyperbolic contours for computing the Bromwich integral, Math. Comp., 76, 259, 1341-1356 (electronic) (2007) · Zbl 1113.65119 · doi:10.1090/S0025-5718-07-01945-X |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
