A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

Liu, Yang; Du, Yanwei; Li, Hong; Li, Jichun; He, Shuping

doi:10.1016/j.camwa.2015.09.012

Cited by 128 publications

(36 citation statements)

References 54 publications

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“…Different fast algorithms, which cover the fast computation of time fractional derivative, fast algorithm of nonlinear problem in time, fast calculation of nonlinear problem in space, fast computation of Matrix and so forth, have different acceleration strategies and features. Jiang et al [36] proposed a fast method of the time Caputo fractional derivative, which can reduce the computing time resulted in by the nonlocality of fractional derivative; Liu et al [25], Liu et al [27], and Yin et al [14] considered the fast calculation for time FPDEs based on the Xu' s two-grid FE methods [26], which can reduce the calculating time yielded by the nonlinear term; Zhao et al [30] developed a fast Hermite FE algorithm to improve the computational efficiency of Matrix, and presented a block circulant preconditioner; Yuste and Quintana-Murillo [28] presented the fast and robust adaptive methods with finite difference scheme for the time fractional diffusion equations; Xu et al [22], Wu and Zhou [23] considered the parareal algorithms for solving the linear time fractional ordinary or partial differential equations (FO(P)DEs), respectively; Zeng et al [8] presented a unified stable fast time-stepping method for fractional derivative and integral operators. Recently, Liu et al [9] proposed a fast TT-M FE algorithm for time fractional water wave model, which is developed to deal with time-consuming problem of nonlinear iteration used in the standard nonlinear Galerkin FE method for nonlinear term.…”

Section: Introductionmentioning

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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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%

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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“…The main idea of this method is using a coarse-grid space to produce a rough approximation of the solution for nonlinear problems, and then use it as the initial guess for one Newton-like iteration on the fine grid. Two-grid discretization method has been widely used for different kinds of problems, such as elliptic Equations [9,10], parabolic equations [12][13][14][15][16][17], eigenvalue problems [18][19][20] stochastic partial differential equations [21] and fractional differential equations [22,23]. The two-grid discretization idea is also used for nonlinear coupled equations, such as the complicated miscible displacement problems [24][25][26] and fluid flow in porous media [27].…”

Section: Introductionmentioning

confidence: 99%

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

Chen

Huang

2019

Numerical Methods Partial

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In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H2 for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.

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“…In the series of works [22][23][24], Ervin and Roop presented a first rigorous analysis for the stationary fractional advection dispersion equation based on a variational formulation. Then the discontinuous Galerkin method [25], mixed finite element method [26][27][28][29][30], Petrov Galerkin method [31] and the least-squared mixed method are proposed [32] for stationary fractional diffusion equations, consecutively.…”

Section: Introductionmentioning

confidence: 99%

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

Yuan

Chen

2018

Adv Differ Equ

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In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings.

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A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

Cited by 128 publications

References 54 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

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

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