L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation

doi:10.1007/s42967-023-00257-x

Abstract

Abstract: In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be α-robust using the newly established Gronwall inequalities, that is, it remains valid when α → 1^-. Numerical experiments are given to demonstrate the theoretical statements.

Key words: Caputo-Hadamard derivative, Discrete Gronwall inequality, L1 formula, Local discontinuous Galerkin (LDG) method, Error estimate

CLC Number:

Zhen Wang. L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 203-227.

TrendMD

References

1. Cai, M., Karniadakis, G.E., Li, C.P.: Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant. Chaos 32(7), 071101 (2022)
2. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71, 455–478 (2002)
3. 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)
4. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
5. Fan, E.Y., Li, C.P., Li, Z.Q.: Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems. Commun. Nonlinear Sci. Numer. Simul. 106, 106096 (2022)
6. Gohar, M., Li, C.P., Li, Z.Q.: Finite difference methods for Caputo-Hadamard fractional differential equations. Mediterr. J. Math. 17, 194 (2020)
7. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1988)
8. Huang, C., Stynes, M.: α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation. Numer. Algorithms 87, 1749–1766 (2021)
9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)
10. Li, C.P., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)
11. Li, C.P., Li, D.X., Wang, Z.: L1/LDG method for the generalized time-fractional Burgers equation. Math. Comput. Simul. 187, 357–378 (2021)
12. Li, C.P., Li, Z.Q.: Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation. J. Nonlinear Sci. 31(2), 31 (2021)
13. Li, C.P., Li, Z.Q.: The blow-up and global existence of solution to Caputo-Hadamard fractional partial differential equation with fractional Laplacian. J. Nonlinear Sci. 31(5), 80 (2021)
14. Li, C.P., Li, Z.Q., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85(2), 41 (2020)
15. Li, C.P., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis. Appl. Numer. Math. 140, 1–22 (2019)
16. Li, C.P., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: mathematical analysis. Appl. Numer. Math. 150, 587–606 (2020)
17. Li, C.P., Wang, Z.: The discontinuous Galerkin finite element method for Caputo-type nonlinear conservation law. Math. Comput. Simul. 169, 51–73 (2020)
18. Li, C.P., Wang, Z.: Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution. Math. Comput. Simul. 182, 838–857 (2021)
19. Li, C.P., Wang, Z.: Numerical methods for the time fractional convection-diffusion-reaction equation. Numer. Funct. Anal. Optim. 42(10), 1115–1153 (2021)
20. Li, C.P., Wang, Z.: L1/local discontinuous Galerkin method for the time-fractional Stokes equation. Numer. Math. Theor. Meth. Appl. 15(4), 1099–1127 (2022)
21. Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86–103 (2018)
22. Li, D., Wang, J., Zhang, J.: Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J. Sci. Comput. 39(6), A3067–A3088 (2017)
23. Liao, H., Li, D., Zhang, J.: Sharp error estimate of nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)
24. Lomnitz, C.: Creep measurements in igneous rocks. J. Geol. 64, 473–479 (1956)
25. Lomnitz, C.: Linear dissipation in solids. J. Appl. Phys. 28, 201–205 (1957)
26. Lomnitz, C.: Application of the logarithmic creep law to stress wave attenuation in the solid earth. J. Geophys. Res. 67(1), 365–367 (1962)
27. Tarasov, V.E.: Entropy interpretation of Hadamard-type fractional operators: fractional cumulative entropy. Entropy 24, 1852 (2022)
28. Wang, Z.: High-order numerical algorithms for the time-fractional convection-diffusion equation. Int. J. Comput. Math. 99(11), 2327–2348 (2022)
29. Wang, Z.: The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem. J. Appl. Math. Comput. 68, 4391–4413 (2022)
30. Wang, Z., Ou, C., Vong, S.: A second-order scheme with nonuniform time grids for Caputo-Hadamard fractional sub-diffusion equations. J. Comput. Appl. Math. 414, 114448 (2022)
31. Yin, B., Liu, Y., Li, H., Zeng, F.: A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations. Appl. Numer. Math. 165, 56–82 (2022)

[1]	Min Cao, Yuan Li. Optimal Error Analysis of Linearized Crank-Nicolson Finite Element Scheme for the Time-Dependent Penetrative Convection Problem [J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 264-288.
[2]	Meiqi Tan, Juan Cheng, Chi-Wang Shu. The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations [J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 115-150.
[3]	Yu Wang, Min Cai. Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions [J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1674-1696.
[4]	Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu. Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1509-1530.
[5]	Ohannes A. Karakashian, Michael M. Wise. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 823-854.
[6]	Andreas Dedner, Tristan Pryer. Discontinuous Galerkin Methods for a Class of Nonvariational Problems [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 634-656.
[7]	Xue Hong, Yinhua Xia. Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 530-562.
[8]	Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.
[9]	Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.
[10]	Leilei Wei, Shuying Zhai, Xindong Zhang. Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 429-444.
[11]	Min Zhang, Yang Liu, Hong Li. High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 613-640.
[12]	Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.
[13]	Yanyong Wang, Yubin Yan, Ye Hu. Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 505-523.