Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes

Li, Meng; Huang, Chengming; Jiang, Fengze

doi:10.1080/00036811.2016.1186271

Cited by 27 publications

(11 citation statements)

References 24 publications

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“…The new schemes admit some advantages for both capturing the initial layer and solving the models with small parameter α. More studies can be found in [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].…”

Section: Introductionmentioning

confidence: 99%

Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay

She

Niu

2021

Journal of Function Spaces

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A linearized numerical scheme is proposed to solve the nonlinear time-fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method, and extrapolation methods in the temporal direction. A novel discrete fractional Grönwall inequality is established. Thanks to the inequality, the error estimate of a fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.

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Section: Introductionmentioning

confidence: 99%

Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay

She

Niu

2021

Journal of Function Spaces

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“…Zhang and Jiang studied a Crank-Nicolson Legendre spectral method [51] for nonlinear time fractional diffusion-wave equations. Since multi-term TFPDEs describe certain diffusion processes more accurately, various numerical methods, including Galerkin FEMs [3,17,50,56], orthogonal spline collocation methods [45], finite difference methods [5,16], compact difference methods [28], spectral methods [47,55], finite volume methods [46] have been developed for such equations. In addition, FEMs [35,36], compact finite difference methods [6,11], finite difference methods [38], Galerkin spectral element methods [6,30], singular boundary methods jointed with dual reciprocity methods [29] are also employed to multi-term TFWEs.…”

Section: Introductionmentioning

confidence: 99%

A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

Fan¹

2021

EAJAM

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An unconditionally stable fully-discrete scheme on regular and anisotropic meshes for multi-term time-fractional mixed diffusion and diffusion-wave equations (TFMDDWEs) with variable coefficients is developed. The approach is based on a nonconforming mixed finite element method (FEM) in space and classical L1 time-stepping method combined with the Crank-Nicolson scheme in time. Then, the unconditionally stability analysis of the fully-discrete scheme is presented. The convergence for the original variable u and the flux p = µ(x)∇u, respectively, in H 1-and L 2-norms is derived by using the relationship between the projection operator R h and the interpolation operator I h. Interpolation postprocessing technique is used to establish superconvergence results. Finally, numerical tests are provided to demonstrate the theoretical analysis.

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“…Jin et al [9] revisited the error analysis of the L1-formula, especially for the nonsmooth initial data, the authors obtained O(τ ) convergence rate. Li et al [12] applied finite element methods based on L1-formula to study multi-term fractional diffusion equations, obtained the unconditional stability and optimal error estimates, and gave some numerical examples with higher spatial dimension. Zhao et al [10] constructed two conforming and nonconforming MFE schemes with L1-formula to solve time fractional diffusion equations, and gave the stability and convergence results.…”

Section: Introductionmentioning

confidence: 99%

Finite Volume Element Methods for Two-Dimensional Time Fractional Reaction-Diffusion Equations on Triangular Grids

Fang

Zhao²,

et al. 2021

Preprint

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In this paper, the time fractional reaction-diffusion equations with the Caputo fractional derivative are solved by using the classical L1-formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability result and optimal a priori error estimate in L 2 (Ω)-norm are derived, but it is difficult to obtain the corresponding results in H 1 (Ω)-norm, so another analysis technique is introduced and used to achieve our goal. Finally, two numerical examples in different spatial dimensions are given to verify the feasibility and effectiveness.

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Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes

Cited by 27 publications

References 24 publications

Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay

Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay

A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

Finite Volume Element Methods for Two-Dimensional Time Fractional Reaction-Diffusion Equations on Triangular Grids

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