Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation

Du, Yanwei; Liu, Yang; Li, Hong; Fang, Zhichao; He, Shuping

doi:10.1016/j.jcp.2017.04.078

Cited by 61 publications

(17 citation statements)

References 59 publications

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“…Theorem 2. Let {U n , Z n } be the solution of the MFVE scheme (13), then there exists a constant C > 0 independent of h and τ such that…”

Section: Existence Uniqueness and Stability Analysis For The Mfve Scmentioning

confidence: 99%

“…This section mainly studies the problem of convergence analysis for the MFVE scheme (13). First, let {ũ h ,σ h } be the generalized MFVE projection of {u, σ} defined by (17), thus the errors are expressed as…”

Section: Convergence Analysis For the Mfve Schemementioning

confidence: 99%

“…Making use of (4), (13) and (14), and applying the generalized MFVE projection, we obtain the error equations…”

Section: Convergence Analysis For the Mfve Schemementioning

confidence: 99%

“…However, there are still many FPDEs for which it is difficult to obtain analytic solutions. Thus, a lot of numerical techniques have been constructed and studied to solve FPDEs [12][13][14][15][16][17][18][19][20][21][22], including finite difference (FD) methods, finite element (FE) methods, mixed finite element (MFE) methods, spectral methods, (local) discontinuous Galerkin (L)DG methods and so on.…”

Section: Introductionmentioning

confidence: 99%

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A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Zhao

Fang

et al. 2019

Mathematics

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In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

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“…Theorem 2. Let {U n , Z n } be the solution of the MFVE scheme (13), then there exists a constant C > 0 independent of h and τ such that…”

Section: Existence Uniqueness and Stability Analysis For The Mfve Scmentioning

confidence: 99%

Section: Convergence Analysis For the Mfve Schemementioning

confidence: 99%

“…Making use of (4), (13) and (14), and applying the generalized MFVE projection, we obtain the error equations…”

Section: Convergence Analysis For the Mfve Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Zhao

Fang

et al. 2019

Mathematics

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“…Given the existing fractional derivatives, the analytical solutions for fractional partial differential equations (FPDEs) are difficult to obtain. Therefore, we have to consider effective numerical methods, including finite element methods (FEMs) , discontinuous Galerkin methods , spectral methods , finite difference methods (FDMs) , collocation methods , finite volume (element) methods (FVMs) , wavelets methods , and meshless methods .…”

Section: Introductionmentioning

confidence: 99%

High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative

Zhang

Liu

2019

Numerical Methods Partial

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In this article, a local discontinuous Galerkin (LDG) method is studied for numerically solving the fractal mobile/immobile transport equation with a new time Caputo-Fabrizio fractional derivative. The stability of the LDG scheme is proven, and a priori error estimates with the second-order temporal convergence rate and the (k + 1)th order spatial convergence rate are derived in detail. Finally, numerical experiments based on P k , k = 0, 1, 2, 3, elements are provided to verify our theoretical results. KEYWORDSa priori error analysis, Caputo-Fabrizio fractional derivative, fractal mobile/immobile transport equation, LDG method, stability 1 Numer Methods Partial Differential Eq. 2019;35:1588-1612. wileyonlinelibrary.com/journal/num © 2019 Wiley Periodicals, Inc. 1588 ] ds, (2.4)and c k ∈ (t k − 1 , t k ). By taking z(t) = u(x, t) in (2.2) at t = t n , we can obtain CF 0 t u(x, t n ) =

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Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

Yue

Liu

Shu

et al. 2019

Math Methods in App Sciences

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In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain Ω. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the L 2 (Ω) norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.Another important note is that no matter which fully discrete scheme is utilized, there always exists the computational challenge in nonlocality caused by fractional differential operators [27]. 15 Quite many scholars are working to identify algorithms most appropriate to overcome the challenge and utilize computer resources. Multigrid exploits a hierarchy of grids or multiscale representations, and reduces the error of the approximation at a number of frequencies (from global smooth to local oscillation) simultaneously [28-31], which makes multigrid as an extremely superior solver or preconditioner of particular interest. Pang and Sun presented an efficient V-cycle geometric multi-20 grid (GMG) with fast Fourier transform (FFT) to solve one-dimensional space-fractional diffusion equations (SFDEs) discretized by an implicit FD scheme [32]. Bu et al. extended and analyzed the V-cycle GMG for one-dimensional MTFADEs via the FD in temporal and FE in spatial directions [26]. Zhou and Wu discussed the FE F-cycle GMG to linear stationary FADEs in Riemann-Liouvlle

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Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation

Cited by 61 publications

References 59 publications

A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative

Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

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