Abstract
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.
Highlights
Fractional differential equations are generalized differential operations from integer orders to fractional derivative operations
A finite difference scheme is employed in time and a Petrov-Galerkin finite element method is employed in space
The time-fractional differential equation is obtained by replacing the integer order in the classical model to fractional derivative, the fractional derivative at a certain time depends on all the values of the function before this time point, so the fractional partial differential equation is applicable for problems with memory process, genetic property and heterogeneous material (Rossikhin and Shitikova, 1997; Ichise et al, 1971)
Summary
Fractional differential equations are generalized differential operations from integer orders to fractional derivative operations. In order to simulate problems with irregular convex domains, Fan et al (Fan et al, 2017) proposed an unstructured mesh finite element method to solve the time-space fractional wave equation on an irregular convex domain. (Ying and Henriquez, 2007) is a kernel-free boundary integral (KFBI) method for solving elliptic BVPs. In (Hou and Liu, 2005), a finite element method with non-body-fitting grids is proposed to solve elliptic equations with matrix coefficients and sharp-edged interfaces. We propose a Petrov-Galerkin finite element method to solve the time-fractional diffusion equation with interfaces.
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