Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

Zhao, Jingjun; Xiao, Jingyu; Xu, Yang

doi:10.1155/2013/857205

Cited by 8 publications

(6 citation statements)

References 34 publications

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“…Numerical methods for the general multi-term case for an ordinary differential equation were considered in [15,6]. In [36], a scheme based on the finite element method in space and a specialized finite difference method in time was proposed for (1.1), and error estimates were derived. We also refer to [22] for a numerical scheme based on a fractional predictorcorrector method for the multi-term time fractional wave-diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

The Galerkin finite element method for a multi-term time-fractional diffusion equation

Jin

Lazarov

Liu

et al. 2015

Journal of Computational Physics

233

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We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.2 BANGTI JIN, RAYTCHO LAZAROV, YIKAN LIU, AND ZHI ZHOU space and slow asymptotic decay in time [30], which in turn also impacts related numerical analysis [12] and inverse problems [14,30].The model (1.1) was developed to improve the modeling accuracy of the single-term model (1.3) for describing anomalous diffusion. For example, in [31], a two-term fractionalorder diffusion model was proposed for the total concentration in solute transport, in order to distinguish explicitly the mobile and immobile status of the solute using fractional dynamics. The kinetic equation with two fractional derivatives of different orders appears also quite naturally when describing subdiffusive motion in velocity fields [26]; see also [16] for discussions on the model for wave-type phenomena.There are very few mathematical studies on the model (1.1). Luchko [23] established a maximum principle for problem (1.1), and constructed a generalized solution for the case f ≡ 0 using the multinomial Mittag-Leffler function. Jiang et al [9] derived analytical solutions for the diffusion equation with fractional derivatives in both time and space. Li and Yamamoto [20] established existence, uniqueness, and the Hölder regularity of the solution using a fixed point argument for problem (1.1) with variable coefficients {bi}. Very recently, Li et al [19] showed the uniqueness and continuous dependence of the solution on the initial value v and the source term f , by exploiting refined properties of the multinomial Mittag-Leffler function.The applications of the model (1.1) motivate the design and analysis of numerical schemes that have optimal (with respect to data regularity) convergence rates. Such schemes are especially valuable for problems where the solution has low regularity. The case m = 0, i.e., the single-term model (1.3), has been extensively studied, and stability and error estimates were provided; see [21,35] for the finite difference method, [18,34] for the spectral method, [25,27,28,12,11,10] for the finite element method, and [3,7] for meshfree methods based on radial basis functions, to name a few. In particular, in [10,11,12], the authors established almost optimal error estimates with respect to the regularity of the initial data v and the right hand side f for a semidiscrete Galerkin scheme. These studies includ...

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Section: Introductionmentioning

confidence: 99%

The Galerkin finite element method for a multi-term time-fractional diffusion equation

Jin

Lazarov

Liu

et al. 2015

Journal of Computational Physics

233

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“…In addition, finding the reliable and powerful analytical and numerical methods for solving FPDEs has been focused in the last two decades. According to the some mathematical literatures, FPDEs have been progressed in various problems in science and engineering such as the Schröinger, diffusion and telegraph fractional equations [10][11][12]. Furthermore, many scholars and researchers are focused on the numerical methods of FPDEs in the last two decades.…”

Section: Introductionmentioning

confidence: 99%

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

Yao

2019

Numerical Methods Partial

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In this paper, we consider the numerical solution of the time‐fractional telegraph equation with a nonlocal boundary condition. A novel barycentric Lagrange interpolation collocation method is developed to solve this equation. Two difficulties have been sorted: the singularity of the integration and the higher accuracy. At the same, we put forward a steady barycentric Lagrange interpolation technique to overcome the new “Runge” phenomenon in computation. Error estimates of the barycentric Lagrange interpolation and the time‐fractional telegraph system for the present method are presented in Sobolev spaces. High convergence rates of the proposed method are obtained and are consisted with the numerical values. Especially in the time dimension, we get the error bound, Ohnt−2 for h‐refinement and Oitaliceht/2nt−2nt−2 for nt‐density in the L2 norms. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time‐fractional telegraph equation. Experiments demonstrate the high convergence rates of the proposed method are consisted with the theoretical values.

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“…As a counterpart of traditional integer order differential equation, fractional differential equation can be obtained by replacing the integer order derivatives with fractional ones in integer order differential equation. Fractional partial differential equations(FPDEs), particularly space and time-fractional equations, have been widely studied to construct the existence of solution and validity of these problems [6][7][8]. In addition, the reliable and powerful numerical and analytical methods for solving FPDEs has been focused in the last two decades.…”

Section: Introductionmentioning

confidence: 99%

A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

Sun¹

2019

AAMM

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In this paper, we firstly present a novel simple method based on a Picard integral type formulation for the nonlinear multi-dimensional variable coefficient fourthorder advection-dispersion equation with the time fractional derivative order α∈(1,2). A new unknown function v(x,t) = ∂u(x,t)/∂t is introduced and u(x,t) is recovered using the trapezoidal formula. As a result of the variable v(x,t) are introduced in each time step, the constraints of traditional plans considering the non-integer time situation of u(x,t) is no longer considered. The stability and solvability are proved with detailed proofs and the precise describe of error estimates is derived. Further, Chebyshev spectral collocation method supports accurate and efficient variable coefficient model with variable coefficients. Several numerical results are obtained and analyzed in multi-dimensional spatial domains and numerical convergence order are consistent with the theoretical value 3−α order for different α under infinite norm.

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Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

Cited by 8 publications

References 34 publications

The Galerkin finite element method for a multi-term time-fractional diffusion equation

The Galerkin finite element method for a multi-term time-fractional diffusion equation

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

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