Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations

Bu, Weiping; Tang, Yifa; Wu, Yingchuan; Yang, Jiayi

doi:10.1016/j.jcp.2014.06.031

Cited by 126 publications

(51 citation statements)

References 25 publications

(37 reference statements)

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“…Ervin and Roop [4] developed the variational solution for spatial fractional advection dispersion equations. Deng [15], Feng et al [17], Bu et al [2], Fan et al [18], Zhao et al [20], Li et al [21], Yue et al [29], Zhang et al [33], Zhu et al [34], Zheng et al [35], Dehghan and Abbaszadeh [37], Chen and Wang [38], Jin et al [41], Li et al [31] considered finite element methods for some space or space-time FPDEs. Heydari [44] [19] used a Crank-Nicolson finite difference methods for space fractional Allen-Cahn equations.…”

Section: Introductionmentioning

confidence: 99%

“…See[2]) Let s and r be real numbers satisfying 0 < r ≤ k+1, 0 ≤ s < r. Then there exist a projector Π h and a positive constant C depending only on Ω such that, for any function u ∈ H s (Ω), the following estimate holdsu − Π h u H s (Ω) ≤ Ch r−s u H r (Ω) . (5.2) By Lemma 5.2, the operator P h defined in (5.1) has the following estimate property with respect to the seminorm | · | µ .…”

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confidence: 99%

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Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

Yin

Liu

et al. 2019

Journal of Computational Physics

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In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth and non-smooth solutions. The implicit second-order θ scheme containing both implicit Crank-Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order θ FE scheme on the time coarse mesh τ c is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh τ < τ c is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved. RL D α c,y u = 1 Fast TT-M FE algorithm combined with second-order θ scheme is used to solve the nonlinear space 1 2 , (2.5) and norm u J β S (Ω) = ( u 2 + |u| 2 J β S (Ω) ) 1 2 , (2.6)and denote by J β S (Ω)(or J β S,0 (Ω)) the closure of C ∞ (Ω)(or C ∞ 0 (Ω)) with respect to · J β S (Ω) .

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

Yin

Liu

et al. 2019

Journal of Computational Physics

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“…Therefore, there have been growing interests recently in developing numerical methods for solving fractional differential equations. Until now, various numerical methods are given for solving fractional differential equations such as finite difference methods [8,19,22,41], spectral methods [18,20], collocation methods [38], finite element methods [5,9], etc.…”

Section: Introductionmentioning

confidence: 99%

Galerkin finite element method for nonlinear fractional Schrödinger equations

2016

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In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

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“…Although there have existed some studies for subdiffusion equation in more than one spatial dimension, say [2,3,12,17,20], A c c e p t e d M a n u s c r i p t the accuracy in temporal direction is low. So there remain a lot of improvements in the present numerical schemes.…”

Section: Introductionmentioning

confidence: 99%

“…In order to overcome this drawback, we constructed the ADI scheme for equation (1). If β → 1, then the derived ADI scheme for (1) is just the classical ADI scheme for equation (2).…”

Section: Introductionmentioning

confidence: 99%

A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions

Chen

2015

International Journal of Computer Mathematics

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In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the β → 1, where 1 − β (∈ (0, 1)) is the order of the Riemann-Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

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Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations

Cited by 126 publications

References 25 publications

Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

Galerkin finite element method for nonlinear fractional Schrödinger equations

A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions

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