An Indirect Finite Element Method for Variable-Coefficient Space-Fractional Diffusion Equations and Its Optimal-Order Error Estimates

Abstract

We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions u(x) to the proposed variable coefficient models in terms of v(x), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations $v_h(x)$ to v(x) and then obtain the approximations $u_h(x)$ of u(x) by plugging $v_h(x)$ into the representation of u(x). Optimal-order convergence estimates of $u(x)-u_h(x)$ are proved in both $L^2$ and $H^{\alpha /2}$ norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.

Funding

OSD/ARO(MURI Grant W911NF-15-1-0562); National Science Foundation(Grant DMS-1620194)