scholarly journals Leapfrog/Finite Element Method for Fractional Diffusion Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhengang Zhao ◽  
Yunying Zheng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Element Analysis ◽  
Bounded Interval ◽  
Fully Discrete ◽  
Leapfrog Scheme ◽  
Element Method

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove anL2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.

Application of low-dimensional finite element method to fractional diffusion equation

2014 ◽  
Vol 05 (04) ◽  
pp. 1450022 ◽  
Author(s):  
Jincun Liu ◽  
Hong Li ◽  
Zhichao Fang ◽  
Yang Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Fractional Differential Equations ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Fractional Differential ◽  
Low Dimensional ◽  
Element Method

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

scholarly journals A Petrov-Galerkin Finite Element Method for Solving the Time-fractional Diffusion Equation with Interface

2018 ◽  
Vol 10 (4) ◽  
pp. 136
Author(s):  
Liwei Shi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Discontinuous Coefficients ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Time Fractional Diffusion Equation ◽  
Element Method

Time-fractional partial differential equation is widely applied in a variety of disciplines, its numerical solution has attracted much attention from researchers in recent years. Time-fractional differential equations with interfaces is a more challenging problem because the governing equation has discontinuous coefficients at interfaces and sometimes singular source term exists. In this paper, we propose a Petrov-Galerkin finite element method for solving the two-dimensional time-fractional diffusion equation with interfaces. In this method, a finite difference scheme is employed in time and a Petrov-Galerkin finite element method is employed in space. Extensive numerical experiments show that for a fractional diffusion equation of order $\alpha$ with interfaces, our method gets to $(2-\alpha)$-order accurate in the $L^2$ and $L^{\infty}$ norm.

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