<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.gif" display="inline" overflow="scroll"><mml:mi>θ</mml:mi></mml:math>schemes for finite element discretization of the space–time fractional diffusion equations

Guan, Qingguang; Gunzburger, Max

doi:10.1016/j.cam.2015.04.032

Cited by 20 publications

(18 citation statements)

References 24 publications

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“…We first quote the following results.Lemma (). Recall the Cauchy principal value defined by

ψ (x, t) = p . v . \int_{x - δ}^{x + δ} \frac{u false(x, t false) - u false(y, t false)}{| x - y |^{1 + s}} d y = \lim_{ϵ \to 0} \int_{Ω_{ϵ}} \frac{u false(x, t false) - u false(y, t false)}{| x - y |^{1 + s}} d y, 0 < s < 1,

where

Ω_{ϵ} = (x - δ, x + δ) ∖ (x - ϵ, x + ϵ)

.…”

Section: Finite Element Discretizationsunclassified

“…Space‐time fractional diffusion and diffusion‐wave equations are used as models for anomalous transport in many disciplines such as hydrogeology, biology, and so forth. In this article, we consider the following one dimension space‐time fractional diffusion‐wave equation (see ):

true{\begin{array}{l} left & \frac{\partial u}{\partial t} + V \frac{\partial^{γ} u}{\partial t^{γ}} = - c^{2} ℒ u + f (x, t), x \in Ω, 1 < γ < 2, \\ left & u (x, t) = 0, x \in Γ, t \in [0, T], \\ left & u (x, 0) = u_{0} (x), x \in Ω, u_{t} (x, 0) = ψ (x), x \in Ω, \end{array}

where

scriptL

denotes the following space‐fractional operator

L u = \int_{x - δ}^{x + δ} \frac{u false(x, t false) - u false(y, t false)}{| x - y |^{1 + s}} d y, 0 < s < 1,

and the constant

V \geq 0

. Specially, when

V = 0

, the Eq.…”

Section: Introductionsmentioning

confidence: 99%

“…The integral in

L u

is divergent in the classic Riemann sense, and should be understood in the Hadamard finite part sense. These types of integrals are often referred to as “Hadamard finite part integrals” or “hypersingular integrals.” From , we know that

δ > 0

. What is more, with

δ = \infty

, it is well known that

L = {(- Δ)}^{s}

, the fractional Laplacian operator.…”

Section: Introductionsmentioning

confidence: 99%

“…For nonsteady case, Guan and Max analyzed time‐stepping methods and considered the stability and convergence. For a space‐time fractional diffusion equation, Guan and Max studied

θ

‐schemes for piece‐linear finite element discretization and gave convergence rate

O (τ + h^{2})

.…”

Section: Introductionsmentioning

confidence: 99%

“…Space-time fractional diffusion and diffusion-wave equations are used as models for anomalous transport in many disciplines such as hydrogeology, biology, and so forth. In this article, we consider the following one dimension space-time fractional diffusion-wave equation (see [19]):…”

Section: Introductionsmentioning

confidence: 99%

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A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

Liu

Cheng

2017

Numerical Methods Partial

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In this article, we study fast discontinuous Galerkin finite element methods to solve a space‐time fractional diffusion‐wave equation. We introduce a piecewise‐constant discontinuous finite element method for solving this problem and derive optimal error estimates. Importantly, a fast solution technique to accelerate Toeplitz matrix‐vector multiplications which arise from discontinuous Galerkin finite element discretization is developed. This fast solution technique is based on fast Fourier transform and it depends on the special structure of coefficient matrices. In each temporal step, it helps to reduce the computational work from O ( N 3 ) required by the traditional methods to O ( N log 2 N ) , where N is the size of the coefficient matrices (number of spatial grid points). Moreover, the applicability and accuracy of the method are verified by numerical experiments including both continuous and discontinuous examples to support our theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2043–2061, 2017

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“…We first quote the following results.Lemma (). Recall the Cauchy principal value defined by

ψ (x, t) = p . v . \int_{x - δ}^{x + δ} \frac{u false(x, t false) - u false(y, t false)}{| x - y |^{1 + s}} d y = \lim_{ϵ \to 0} \int_{Ω_{ϵ}} \frac{u false(x, t false) - u false(y, t false)}{| x - y |^{1 + s}} d y, 0 < s < 1,

where

Ω_{ϵ} = (x - δ, x + δ) ∖ (x - ϵ, x + ϵ)

.…”

Section: Finite Element Discretizationsunclassified

true{\begin{array}{l} left & \frac{\partial u}{\partial t} + V \frac{\partial^{γ} u}{\partial t^{γ}} = - c^{2} ℒ u + f (x, t), x \in Ω, 1 < γ < 2, \\ left & u (x, t) = 0, x \in Γ, t \in [0, T], \\ left & u (x, 0) = u_{0} (x), x \in Ω, u_{t} (x, 0) = ψ (x), x \in Ω, \end{array}

where

scriptL

denotes the following space‐fractional operator

L u = \int_{x - δ}^{x + δ} \frac{u false(x, t false) - u false(y, t false)}{| x - y |^{1 + s}} d y, 0 < s < 1,

and the constant

V \geq 0

. Specially, when

V = 0

, the Eq.…”

Section: Introductionsmentioning

confidence: 99%

“…The integral in

L u

δ > 0

. What is more, with

δ = \infty

, it is well known that

L = {(- Δ)}^{s}

, the fractional Laplacian operator.…”

Section: Introductionsmentioning

confidence: 99%

“…For nonsteady case, Guan and Max analyzed time‐stepping methods and considered the stability and convergence. For a space‐time fractional diffusion equation, Guan and Max studied

θ

‐schemes for piece‐linear finite element discretization and gave convergence rate

O (τ + h^{2})

.…”

Section: Introductionsmentioning

confidence: 99%

Section: Introductionsmentioning

confidence: 99%

See 3 more Smart Citations

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

Liu

Cheng

2017

Numerical Methods Partial

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Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

Sun

Zhang

et al. 2020

Math Methods in App Sciences

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Collocation method for one dimensional nonlocal diffusion equations

Guan

Gunzburger

Zhang

2021

Numerical Methods Partial

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In this paper, the collocation method for solving one dimensional steady state and time dependent nonlocal diffusion equations is analyzed. The difficulty of applying collocation method to nonlocal diffusion equations comes from the singularity of the kernel. If s < 1 2 the kernel is weakly singular, however, if s ≥ 1 2 the kernel is not integrable in Riemann sense. So that the Hadamard finite part integral is introduced to overcome this difficulty. For analysis and performance, a "balance" term is added to discretize the nonlocal operator. Numerical results validate the theorems.

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θschemes for finite element discretization of the space–time fractional diffusion equations

Cited by 20 publications

References 24 publications

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

Collocation method for one dimensional nonlocal diffusion equations

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