Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain

Zhao, Yanxing; Wang, Fenling; Hu, Xiaohan; Shi, Zhengguang; Tang, Yifa

doi:10.1016/j.camwa.2018.11.028

Cited by 15 publications

(13 citation statements)

References 39 publications

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“…Although there is a vast literature on approximation approaches to TFDEs and TFWEs, numerical methods for TFMDDWEs are less developed. Nevertheless, we can note spectral methods [4,24], FEMs [8,44,49], implicit difference methods [32]. In particular, Liu et al [24] constructed an ADI spectral scheme for two-dimensional multi-term TFMDDWEs using Legendre spectral approximation in space and finite difference discretisation in time.…”

Section: Introductionmentioning

confidence: 99%

“…The regular and quasi-uniform assumptions are fundamental conditions for conventional finite elements. However, there are solutions with an essential changes in a certain direction only and in such situations, anisotropic finite elements provide better options [13,33,41,49,53]. This paper presents an unconditionally stable fully-discrete scheme for multiterm TFMDDWEs with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Introductionmentioning

confidence: 99%

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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“…They proved the numerical stability and convergence of the scheme, and also considered the non-smooth solution case by adding some correction terms. Zhao et al [22] established a fully-discrete scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion-wave equations by using linear triangle finite element approximation in space and classical L1 approximation combined with the Crank-Nicolson technique in time. Then, the unconditional stability and the global superconvergence of the proposed scheme were proved.…”

Section: Introductionmentioning

confidence: 99%

A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity

Huang

Zhang

Arshad

et al. 2021

Numerical Methods Partial

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In this paper, a superlinear convergence scheme for the multi-term and distribution-order fractional wave equation with initial singularity is proposed. The initial singularity of the solution of the multi-term time fractional partial differential equation often generate a singular source, it increases the difficulty to numerically solve the equation. Thus, after discretizing the spatial distribution-order derivative by the midpoint quadrature, an integral transformation is applied to deal with the temporal direction for obtaining a temporal superlinear convergence scheme based on the uniform mesh. Then, the fully discrete scheme is constructed by using Crank-Nicolson technique and L1 approximation in time, and fractional centered difference approximation in space. The convergence and stability of the proposed scheme are rigorously analyzed. Finally, numerical experiments are presented to support the theoretical results of our scheme.

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“…Reference [24] not only derived the analytical solution for a two‐dimensional multi‐term model based on separation of variables and properties of the multivariate Mittag‐Leffler function, but also analyzed the stability and convergence for the implicit difference scheme. Reference [39] handled a 2D multi‐term case with variable coefficient by a FEM and some high accuracy results were deduced. However, it is not rich about the numerical analysis of the time‐fractional multi‐term mixed sub‐diffusion and diffusion‐wave equation.…”

Section: Introductionmentioning

confidence: 99%

“…Equation (1) is very different from the models in [5, 24, 29, 39], because it has a special coupled fractional operator

D_{t}^{α} false(\nabla \cdot false(ω_{5} false(boldx false) \nabla u false(x, t false) false) false)

. When the coefficient

ω_{5} false(boldx false)

equals a constant, Equation (1) can be simplified as

{false\sum}_{r = 1}^{L_{1}} ω_{1 r} D_{t}^{β_{r}} u false(x, t false) + ω_{2} u_{t} false(x, t false) + {false\sum}_{s = 1}^{L_{2}} ω_{3 s} D_{t}^{γ_{s}} u false(x, t false) + ω_{4} u false(x, t false) = ω_{5} Δ u false(x, t false) + ω_{6} D_{t}^{α} false(Δ u false(x, t false) false) + f false(x, t false) .

…”

Section: Introductionmentioning

confidence: 99%

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Shi

Zhao

Wang

et al. 2021

Numerical Methods Partial

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A fully discrete scheme is proposed for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank-Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time-space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.

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Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain

Cited by 15 publications

References 39 publications

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

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

A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

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