L1/finite element methods for time-fractional Keller-Segel equations with weak singularity solution

Jia Xie; Qingfeng Li; Jia Xie; Qingfeng Li

doi:10.3934/era.2026092

Electronic Research Archive

2026, Volume 34, Issue 3: 2066-2086. doi: 10.3934/era.2026092

Previous Article Next Article

Research article

L1/finite element methods for time-fractional Keller-Segel equations with weak singularity solution

Jia Xie ¹,
Qingfeng Li ^{2
,
,}

1.
School of Educational Science, Gannan Normal University, Ganzhou 341000, China
2.
School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China

Received: 29 December 2025 Revised: 23 February 2026 Accepted: 27 February 2026 Published: 06 March 2026

In this paper, we consider the L1/ finite element methods for solving time-fractional Keller-Segel equations with a singular solution. For time direction, the L1 scheme under the nonuniform mesh is considered to approximate the Caputo derivative to handle the singularity of the solution. To further reduce computational storage requirements, we consider a fast L1 scheme based on the sum-of-exponentials skill for calculating fractional derivatives. Then, we derive a fully implicit discrete scheme by combining the finite element method in the spatial direction. Subsequently, unconditional stability and $ \alpha $-robust error estimates of the fully discrete scheme are derived. Finally, numerical examples are presented to verify our theory.
- time-fractional Keller-Segel equations,
- L1 scheme,
- finite element method,
- unconditional stability,
- $ \alpha $-robust error estimates
Citation: Jia Xie, Qingfeng Li. L1/finite element methods for time-fractional Keller-Segel equations with weak singularity solution[J]. Electronic Research Archive, 2026, 34(3): 2066-2086. doi: 10.3934/era.2026092

Related Papers:

Abstract

In this paper, we consider the L1/ finite element methods for solving time-fractional Keller-Segel equations with a singular solution. For time direction, the L1 scheme under the nonuniform mesh is considered to approximate the Caputo derivative to handle the singularity of the solution. To further reduce computational storage requirements, we consider a fast L1 scheme based on the sum-of-exponentials skill for calculating fractional derivatives. Then, we derive a fully implicit discrete scheme by combining the finite element method in the spatial direction. Subsequently, unconditional stability and $ \alpha $-robust error estimates of the fully discrete scheme are derived. Finally, numerical examples are presented to verify our theory.

References

[1]	E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415. https://doi.org/10.1016/0022-5193(70)90092-5 doi: 10.1016/0022-5193(70)90092-5
[2]	E. F. Keller, L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. https://doi.org/10.1016/0022-5193(71)90051-8 doi: 10.1016/0022-5193(71)90051-8
[3]	M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708–729. https://doi.org/10.1016/j.jmaa.2008.07.071 doi: 10.1016/j.jmaa.2008.07.071
[4]	J. Xu, H. Fu, A decoupled linear, mass-conservative block-centered finite difference method for the Keller-Segel chemotaxis system, J. Comput. Phys., 526 (2025), 113775. https://doi.org/10.1016/j.jcp.2025.113775 doi: 10.1016/j.jcp.2025.113775
[5]	J. Hu, X. Zhang, Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations, IMA J. Numer. Anal., 43 (2023), 1450–1484. https://doi.org/10.1093/imanum/drac014 doi: 10.1093/imanum/drac014
[6]	G. Zhou, N. Saito, Finite volume methods for a Keller-Segel system: Discrete energy, error estimates and numerical blow-up analysis, Numer. Math., 135 (2017), 265–311. https://doi.org/10.1007/s00211-016-0793-2 doi: 10.1007/s00211-016-0793-2
[7]	M. Sulman, T. Nguyen, A positivity preserving moving mesh finite element method for the Keller-Segel chemotaxis model, J. Sci. Comput., 80 (2019), 649–666. https://doi.org/10.1007/s10915-019-00951-0 doi: 10.1007/s10915-019-00951-0
[8]	K. Wang, E. Liu, X. Feng, Optimal error estimate of unconditionally positivity-preserving, mass-conserving and energy stable method for the Keller-Segel chemotaxis model, Math. Comput., 94 (2025), 2761–2793. https://doi.org/10.1090/mcom/4041 doi: 10.1090/mcom/4041
[9]	J. Shen, J. Xu, Unconditionally bound preserving and energy dissipative schemes for a class of Keller-Segel equations, SIAM J. Numer. Anal., 58 (2020), 1674–1695. https://doi.org/10.1137/19M1246705 doi: 10.1137/19M1246705
[10]	R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
[11]	F. A. Rihan, Numerical modeling of fractional-order biological systems, Abstr. Appl. Anal., 2013 (2013), 816803. https://doi.org/10.1155/2013/816803 doi: 10.1155/2013/816803
[12]	L. Li, J. Liu, Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal., 50 (2018), 3963–3995. https://doi.org/10.1137/17M1145549 doi: 10.1137/17M1145549
[13]	Y. Zhou, J. Manimaran, L. Shangerganesh, A. Debbouche, Weakness and Mittag-Leffler stability of solutions for time-fractional Keller-Segel models, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 753–761. https://doi.org/10.1515/ijnsns-2018-0035 doi: 10.1515/ijnsns-2018-0035
[14]	M. Bezerra, C. Cuevas, C. Silva, H. Soto, On the fractional doubly parabolic Keller-Segel system modelling chemotaxis, Sci. China Math., 65 (2022), 1827–1874. https://doi.org/10.1007/s11425-020-1846-x doi: 10.1007/s11425-020-1846-x
[15]	M. Costa, C. Cuevas, C. Silva, H. Soto, Well-posedness and blow-up of the fractional Keller-Segel model on domains, Math. Nachr., 296 (2023), 5569–5592. https://doi.org/10.1002/mana.202200235 doi: 10.1002/mana.202200235
[16]	A. Slimani, S. Lakhlifa, A. Guesmia, Analytical solution of one-dimensional Keller-Segel equations via new homotopy perturbation method, Contemp. Math., 5 (2024), 1093–1109. https://doi.org/10.37256/CM.5120242604 doi: 10.37256/CM.5120242604
[17]	M. Zayernouri, A. Matzavinos, Fractional Adams-Bashforth/Moulton methods: An application to the fractional Keller-Segel chemotaxis system, J. Comput. Phys., 317 (2016), 1–14. https://doi.org/10.1016/j.jcp.2016.04.041 doi: 10.1016/j.jcp.2016.04.041
[18]	K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426–447. https://doi.org/10.1016/j.jmaa.2011.04.058 doi: 10.1016/j.jmaa.2011.04.058
[19]	M. Stynes, E. O'Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
[20]	B. Jin, B. Li, Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1–23. https://doi.org/10.1137/16M1089320 doi: 10.1137/16M1089320
[21]	H. Liao, D. Li, J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1112–1133. https://doi.org/10.1137/17M1131829 doi: 10.1137/17M1131829
[22]	H. Liao, W. McLean, J. Zhang, A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57 (2019), 218–237. https://doi.org/10.1137/16M1175742 doi: 10.1137/16M1175742
[23]	N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, Math. Comput., 88 (2019), 2135–2155. https://doi.org/10.1090/mcom/3410 doi: 10.1090/mcom/3410
[24]	N. Kopteva, Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem, Math. Comput., 90 (2021), 19–40. https://doi.org/10.1090/mcom/3552 doi: 10.1090/mcom/3552
[25]	H. Chen, M. Stynes, Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput., 79 (2019), 624–647. https://doi.org/10.1007/s10915-018-0863-y doi: 10.1007/s10915-018-0863-y
[26]	S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
[27]	H. Liao, Y. Yan, J. Zhang, Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations, J. Sci. Comput., 80 (2019), 1–25. https://doi.org/10.1007/s10915-019-00927-0 doi: 10.1007/s10915-019-00927-0
[28]	C. Huang, M. Stynes, A sharp $\alpha$-robust L$^{\infty}(H^{1})$ error bound for a time-fractional Allen-Cahn problem discretised by the Alikhanov L2-1$_\sigma$ scheme and a standard FEM, J. Sci. Comput., 91 (2022), 43. https://doi.org/10.1007/s10915-022-01810-1 doi: 10.1007/s10915-022-01810-1
[29]	H. Chen, M. Stynes, Blow-up of error estimates in time-fractional initial-boundary value problems, IMA J. Numer. Anal., 41 (2021), 974–997. https://doi.org/10.1093/imanum/draa015 doi: 10.1093/imanum/draa015
[30]	Z. Sheng, Y. Liu, Y. Li, Finite element method combined with time graded meshes for the time-fractional coupled Burgers' equations, J. Appl. Math. Comput., 70 (2024), 513–533. https://doi.org/10.1007/s12190-023-01969-2 doi: 10.1007/s12190-023-01969-2
[31]	C. Huang, M. Stynes, Optimal spatial $H^1$-norm analysis of a finite element method for a time-fractional diffusion equation, J. Comput. Appl. Math., 367 (2020), 112435. https://doi.org/10.1016/j.cam.2019.112435 doi: 10.1016/j.cam.2019.112435
[32]	C. Huang, M. Stynes, $\alpha$-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation, Numer. Algorithms, 87 (2021), 1749–1766. https://doi.org/10.1007/s11075-020-01036-y doi: 10.1007/s11075-020-01036-y
[33]	Z. Tan, $\alpha$-robust analysis of fast and novel two-grid fem with nonuniform L1 scheme for semilinear time-fractional variable coefficient diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 131 (2024), 107830. https://doi.org/10.1016/j.cnsns.2024.107830 doi: 10.1016/j.cnsns.2024.107830
[34]	V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, $ 2^{nd}$ edition, Springer, 2006. https://doi.org/10.1007/3-540-33122-0
[35]	N. Liu, H. Qin, Y. Yang, Unconditionally optimal $H^1$-norm error estimates of a fast and linearized galerkin method for nonlinear subdiffusion equations, Comput. Math. Appl., 107 (2022), 70–81. https://doi.org/10.1016/j.camwa.2021.12.012 doi: 10.1016/j.camwa.2021.12.012

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)