Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation

2020 ◽  
Vol 151 ◽  
pp. 141-160 ◽  
Author(s):  
Meng Li ◽  
Dongyang Shi ◽  
Lifang Pei
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Finite Element Methods ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Fractional Diffusion Equation ◽  
Convergence And Superconvergence

scholarly journals A Directly Numerical Algorithm for a Backward Time-Fractional Diffusion Equation Based on the Finite Element Method

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zhousheng Ruan ◽  
Zewen Wang ◽  
Wen Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Algorithm ◽  
Optimization Problem ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Finite Element Method ◽  
Time Fractional Diffusion Equation ◽  
Regularized Optimization

We study a backward problem for a time-fractional diffusion equation, which is formulated into a regularized optimization problem. After solving a sequence of well-posed direct problems by the finite element method, a directly numerical algorithm is proposed for solving the regularized optimization problem. In order to obtain a reasonable regularization solution, we utilize the discrepancy principle with decreasing geometric sequence to choose regularization parameters. One- and two-dimensional examples are given to verify the efficiency and stability of the proposed method.

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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