Error Estimate of Finite Element Approximation for Two-Sided Space-Fractional Evolution Equation with Variable Coefficient

In this paper, we develop and analyze a finite element method (FEM) for a one-dimensional two-sided time-dependent space-fractional diffusion equation (sFDE) with variable diffusivity. We first prove the regularity of the solution to the steady-state sFDE equipped with variable diffusivity and then utilize the regularity estimate to analyze the property of an elliptic projection, which is of great importance to derive the error estimate for the time-dependent evolution problems. It has been well-known that the bilinear form corresponding to the finite element weak formulation for variable coefficient problems may not be coercive, and thus the well-posedness of the weak formulation can not be guaranteed. To perform the error estimate, we reformulate the steady-state equation to its equivalent form and then prove the weak coercivity of the corresponding bilinear form via the Garding's inequality, which leads to optimal-order error estimates of the FEM to the steady-state equation and thus to the interested evolution equation. Compared with some existing works on the FEM to variable-coefficient space-fractional problems, the main advantages of the developed numerical analysis techniques lie in the relaxed assumptions on the variable coefficient and its potential extensions to high-dimensional problems. Numerical experiments are performed to verify the theoretical findings.