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

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.