An $$H^1$$ H 1 -Galerkin mixed finite element method for time fractional reaction–diffusion equation

Liu, Yang; Du, Yanwei; Li, Hong; Wang, Jinfeng

doi:10.1007/s12190-014-0764-7

Cited by 44 publications

(17 citation statements)

References 38 publications

(48 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For = [α, β], we assume the midpoint β+α 2 is a point of discontinuity. We can choose suitable N in partition (7) to make a grid point locate at the point of discontinuity. From [26], we know jump discontinuities belong to fractional Soblev spaces H s ( ) with s ∈ (0, 1).…”

Section: A Piecewise-constant Discontinuous Galerkin Formulationmentioning

confidence: 99%

“…The fractional models can be classified into two principal kinds: space-fractional differential equation and time-fractional one. Numerical methods and theory of solutions to problems for fractional differential equations have been studied extensively by many researchers which mainly cover finite element methods [1][2][3][4], mixed finite element methods [5][6][7], finite difference methods [8][9][10][11][12][13], finite volume (element) methods [14,15], (local) discontinuous Galerkin (L)DG methods [16], spectral methods [17,18] and so forth.…”

Section: Introductionsmentioning

confidence: 99%

See 1 more Smart Citation

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

Liu¹,

Cheng²,

Li³

2017

Numer. Methods Partial Differential Eq.

View full text Add to dashboard Buy / Rent full text

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

show abstract

Section: A Piecewise-constant Discontinuous Galerkin Formulationmentioning

confidence: 99%

Section: Introductionsmentioning

confidence: 99%

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

Liu¹,

Cheng²,

Li³

2017

Numer. Methods Partial Differential Eq.

View full text Add to dashboard Buy / Rent full text

show abstract

“…Another major challenge in solving TFCDEs is improving the convergence order. At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method . Wang and Jia adopted fast finite difference methods for space‐fractional diffusion equations, and they have dropped the required storage and computational cost from O ( N 2 ) and O ( N 3 ) to O ( N log N ) and O ( N ).…”

Section: Introductionmentioning

confidence: 99%

“…Yao, Sun, and Wu applied fractional alternating direction implicit method for solving a class of fractional subdiffusion equations, which led to a significant improvement in convergence. Liu developed an H 1 − Galerkin mixed finite element method for time fractional reaction–diffusion equations and obtained a convergence order of O (Δ t 2 − α ). Jin obtained an error analysis of semidiscrete finite element methods for time fractional diffusion equations (TFDEs).…”

Section: Introductionmentioning

confidence: 99%

“…Guo, Pu, and Huang [7] proved the existence and uniqueness of the fractional Ginzburg-Landau equation, the fractional QG equation, and the fractional Landau-Lifshitz equation with energy estimate, and Kaikina [8] proved the fractional evolution PDE via constructing Green's function.Another major challenge in solving TFCDEs is improving the convergence order. At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method [9][10][11][12][13][14]. Wang [9] and Jia [10] adopted fast finite difference methods for space-fractional diffusion equations, and they have dropped the required storage and computational cost from O N 2 and O N 3 to O .…”

mentioning

confidence: 99%

See 1 more Smart Citation

A space-time spectral method for one-dimensional time fractional convection diffusion equations

Sun

2016

Math. Meth. Appl. Sci.

View full text Add to dashboard Buy / Rent full text

In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

An $$H^1$$ H 1 -Galerkin mixed finite element method for time fractional reaction–diffusion equation

Cited by 44 publications

References 38 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

A space-time spectral method for one-dimensional time fractional convection diffusion equations

Contact Info

Product

Resources

About