scholarly journals Finite Element Method for a Kind of Two-Dimensional Space-Fractional Diffusion Equation with Its Implementation

2015 ◽  
Vol 05 (02) ◽  
pp. 135-157 ◽  
Author(s):  
Beiping Duan ◽  
Zhoushun Zheng ◽  
Wen Cao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Two Dimensional ◽  
Space Fractional Diffusion Equation ◽  
Element Method

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.

A New Numerical Scheme for the Space Fractional Diffusion Equation

2014 ◽  
Vol 599-601 ◽  
pp. 1305-1308 ◽  
Author(s):  
Jun Ying Cao ◽  
Zi Qiang Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Efficient Method ◽  
Numerical Scheme ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Numerical Examples ◽  
Space Fractional Diffusion Equation

We construct the numerical method of the space fractional diffusion equation in this paper. We propose an efficient method for its numerical solution. This method is based on a finite difference in time and finite element method in space. Convergence of the method is rigorously established. A series of numerical examples are provided to support the theoretical claims.

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