Abstract
In this paper, we will consider the variable-order subdiffusion initial-boundary value problem with weakly singular solutions. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in time, two efficient numerical methods (which we call L1 FEM and Alikhanov FEM) are developed, where the finite element method is used in space. Firstly, an improved error analysis is given for the L1 FEM of Huang and Chen (Appl Math Lett 139:108559, 2023), and the derived error bounds remain valid as \(\alpha (t^*)\rightarrow 1^-\) for \(0\le t^*\le T\). To obtain the \(\alpha \)-robust optimal convergent analysis for Alikhanov FEM, the truncation error of the Alikhanov scheme for the variable-order Caputo derivative and an \(\alpha \)-robust bound on the complementary discrete kernels \(\mathbb {P}_j^{(n)}\) are presented. Combining these two results with an \(\alpha \)-robust discrete fractional Gronwall inequality, the optimal convergent results in \(L^\infty (L^2)\) norm and \(L^\infty (H^1)\) norm are derived. Furthermore, by adopting a simple postprocessing technique of the numerical solution, a higher convergence order in space is obtained. Finally, a numerical example is presented to verify the optimal theoretical convergent result.
Similar content being viewed by others
Data Availability
The datasets generated during the current study are available from the corresponding author on reasonable request.
References
Podlubny, I.: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, volume 198 of Math. Sci. Eng. Academic Press, San Diego, CA (1999)
Garrappa, R., Giusti, A., Mainardi, F.: Variable-order fractional calculus: a change of perspective. Commun. Nonlinear Sci. Numer. Simul. 102, 105904 (2021)
Hajipour, M., Jajarmi, A., Baleanu, D., Sun, H.G.: On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 69, 119–133 (2019)
Tseng, C.-C.: Design of variable and adaptive fractional order fir differentiators. Signal Process. 86(10), 2554–2566 (2006)
Haq, S., Ghafoor, A., Hussain, M.: Numerical solutions of variable order time fractional \((1+1)\)- and \((1+2)\)-dimensional advection dispersion and diffusion models. Appl. Math. Comput. 360, 107–121 (2019)
Yan, G., Sun, H.G.: A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives. Appl. Math. Model. 78, 539–549 (2020)
Ruilian, D., Alikhanov, A.A., Sun, Z.-Z.: Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations. Comput. Math. Appl. 79(10), 2952–2972 (2020)
Wei, L., Zhai, S., Zhang, X.: Error estimate of a fully discrete local discontinuous Galerkin method for variable-order time-fractional diffusion equations. Commun. Appl. Math. Comput. 3(3), 429–443 (2021)
Zhang, J.-L., Fang, Z.-W., Sun, H.-W.: Exponential-sum-approximation technique for variable-order time-fractional diffusion equations. J. Appl. Math. Comput. 68(1), 323–347 (2022)
Kheirkhah, F., Hajipour, M., Baleanu, D.: The performance of a numerical scheme on the variable-order time-fractional advection-reaction-subdiffusion equations. Appl. Numer. Math. 178, 25–40 (2022)
Mohsen, Z., George, E.K.: Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 293, 312–338 (2015)
Stynes, M., O’Riordan, E., Gracia, J.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)
Huang, C., Stynes, M., An, N.: Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem. BIT 58(3), 661–690 (2018)
Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comp. 88(319), 2135–2155 (2019)
Ren, J., Huang, C., An, N.: Direct discontinuous Galerkin method for solving nonlinear time fractional diffusion equation with weak singularity solution. Appl. Math. Lett. 102, 106111 (2020)
Zhang, H., Yang, X., Tang, Q., Da, X.: A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation. Comput. Math. Appl. 109, 180–190 (2022)
Zhang, D., An, N., Huang, C.: Local error estimates of the fourth-order compact difference scheme for a time-fractional diffusion-wave equation. Comput. Math. Appl. 142, 283–292 (2023)
Liao, H., McLean, W., Zhang, J.: A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218–237 (2019)
Huang, C., An, N., Xijun, Yu.: A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient. Appl. Numer. Math. 151, 367–379 (2020)
Zhou, B., Chen, X., Li, D.: Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations. J. Sci. Comput. 85(2), 39 (2020)
An, N., Zhao, G., Huang, C., Xijun, Yu.: \(\alpha \)-robust \(H^1\)-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions. Comput. Math. Appl. 118, 159–170 (2022)
Hou, D., Azaïez, M., Chuanju, X.: Müntz spectral method for two-dimensional space-fractional convection-diffusion equation. Commun. Comput. Phys. 26(5), 1415–1443 (2019)
Yang, X., Wu, L., Zhang, H.: A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl. Math. Comput. 457, 128192 (2023)
Zheng, X., Wang, H.: Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J. Numer. Anal. 41(2), 1522–1545 (2021)
Huang, C., Chen, H.: Superconvergence analysis of finite element methods for the variable-order subdiffusion equation with weakly singular solutions. Appl. Math. Lett. 139, 108559 (2023)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, Volume 25 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)
Huang, C., Stynes, M.: Optimal spatial \(H^1\)-norm analysis of a finite element method for a time-fractional diffusion equation. J. Comput. Appl. Math. 367, 112435 (2020)
Chen, H., Stynes, M.: Blow-up of error estimates in time-fractional initial-boundary value problems. IMA J. Numer. Anal. 41(2), 974–997 (2021)
Huang, C., Chen, H., An, N.: \(\beta \)-robust superconvergent analysis of a finite element method for the distributed order time-fractional diffusion equation. J. Sci. Comput. 90(1), 44 (2022)
Huang, C., Stynes, M.: Superconvergence of a finite element method for the multi-term time-fractional diffusion problem. J. Sci. Comput. 82(1), 10 (2020)
Liao, H.-L., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. Commun. Comput. Phys. 30(2), 567–601 (2021)
Chen, H., Wang, Y., Fu, H.: \(\alpha \)-robust \(H^1\)-norm error estimate of nonuniform Alikhanov scheme for fractional sub-diffusion equation. Appl. Math. Lett. 125, 107771 (2022)
Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79(1), 624–647 (2019)
Bramble, J.H., Pasciak, J.E., Steinbach, O.: On the stability of the \(L^2\) projection in \(H^1(\Omega )\). Math. Comp. 71(237), 147–156 (2002)
Huang, C., Stynes, M.: 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(2), 43 (2022)
Acknowledgements
The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China (under Grants 12101360 and 12171278), the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant 2022KJ184, and the Natural Science Foundation of Shandong Province (under Grants ZR2020QA031 and ZR2022MA068). The research of Hu Chen is supported in part by the National Natural Science Foundation of China under Grant 11801026, Natural Science Foundation of Shandong Province under Grant ZR2023MA077, and Fundamental Research Funds for the Central Universities (No. 202264006). The research of Xijun Yu is supported in part by the National Natural Science Foundation of China under Grant 12071046.
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Code Availability
Codes of the current study are available from the authors on reasonable request.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Huang, C., An, N., Chen, H. et al. \(\alpha \)-Robust Error Analysis of Two Nonuniform Schemes for Subdiffusion Equations with Variable-Order Derivatives. J Sci Comput 97, 43 (2023). https://doi.org/10.1007/s10915-023-02357-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-023-02357-5
Keywords
- The subdiffusion equation with variable-order derivatives
- Weak singularity
- The nonuniform L1 scheme
- The nonuniform Alikhanov scheme
- Finite element methods