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

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 $\alpha $-robust using the newly established Gronwall inequalities, that is, it remains valid when $\alpha \rightarrow 1^-$. Numerical experiments are given to demonstrate the theoretical statements.

Keywords

Caputo-Hadamard derivative / Discrete Gronwall inequality / L1 formula / Local discontinuous Galerkin (LDG) method / Error estimate / 65M06 / 65M12 / 65M60

References

[1] Cai M, Karniadakis GE, Li CP. Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant. Chaos. 2022, 32(7): 071101

[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.. 2002, 71: 455-478

[3] Chen H, Stynes M. Blow-up of error estimates in time-fractional initial-boundary value problems. IMA J. Numer. Anal.. 2021, 41(2): 974-997

[4] Ciarlet PG. The Finite Element Method for Elliptic Problems. 1978, Amserdam, North-Holland

[5] Fan EY, Li CP, Li ZQ. Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems. Commun. Nonlinear Sci. Numer. Simul.. 2022, 106: 106096

[6] Gohar M, Li CP, Li ZQ. Finite difference methods for Caputo-Hadamard fractional differential equations. Mediterr. J. Math.. 2020, 17: 194

[7] Hardy GH, Littlewood JE, Pólya G. Inequalities. 1988, Cambridge, Cambridge University Press

[8] Huang C, Stynes M. $\alpha $-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation. Numer. Algorithms. 2021, 87: 1749-1766

[9] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. 2006, Amsterdam, Elsevier Science

[10] Li CP, Cai M. Theory and Numerical Approximations of Fractional Integrals and Derivatives. 2019, Philadelphia, SIAM

[11] Li CP, Li DX, Wang Z. L1/LDG method for the generalized time-fractional Burgers equation. Math. Comput. Simul.. 2021, 187: 357-378

[12] Li CP, Li ZQ. Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation. J. Nonlinear Sci.. 2021, 31(2): 31

[13] Li CP, Li ZQ. The blow-up and global existence of solution to Caputo-Hadamard fractional partial differential equation with fractional Laplacian. J. Nonlinear Sci.. 2021, 31(5): 80

[14] Li CP, Li ZQ, Wang Z. Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput.. 2020, 85(2): 41

[15] Li CP, Wang Z. The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis. Appl. Numer. Math.. 2019, 140: 1-22

[16] Li CP, Wang Z. The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: mathematical analysis. Appl. Numer. Math.. 2020, 150: 587-606

[17] Li CP, Wang Z. The discontinuous Galerkin finite element method for Caputo-type nonlinear conservation law. Math. Comput. Simul.. 2020, 169: 51-73

[18] Li CP, Wang Z. Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution. Math. Comput. Simul.. 2021, 182: 838-857

[19] Li CP, Wang Z. Numerical methods for the time fractional convection-diffusion-reaction equation. Numer. Funct. Anal. Optim.. 2021, 42(10): 1115-1153

[20] Li CP, Wang Z. L1/local discontinuous Galerkin method for the time-fractional Stokes equation. Numer. Math. Theor. Meth. Appl.. 2022, 15(4): 1099-1127

[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.. 2018, 24: 86-103

[22] Li D, Wang J, Zhang J. Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J. Sci. Comput.. 2017, 39(6): A3067-A3088

[23] Liao H, Li D, Zhang J. Sharp error estimate of nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal.. 2018, 56: 1112-1133

[24] Lomnitz C. Creep measurements in igneous rocks. J. Geol.. 1956, 64: 473-479

[25] Lomnitz C. Linear dissipation in solids. J. Appl. Phys.. 1957, 28: 201-205

[26] Lomnitz C. Application of the logarithmic creep law to stress wave attenuation in the solid earth. J. Geophys. Res.. 1962, 67(1): 365-367

[27] Tarasov VE. Entropy interpretation of Hadamard-type fractional operators: fractional cumulative entropy. Entropy. 2022, 24: 1852

[28] Wang Z. High-order numerical algorithms for the time-fractional convection-diffusion equation. Int. J. Comput. Math.. 2022, 99(11): 2327-2348

[29] Wang Z. The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem. J. Appl. Math. Comput.. 2022, 68: 4391-4413

[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.. 2022, 414: 114448

[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.. 2022, 165: 56-82

Funding

National Natural Science Foundation of China(12101266)

RIGHTS & PERMISSIONS

Shanghai University