Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations

Abstract

Optimal-order error estimates in the energy norm and the $L^2$ norm were previously proved in the literature for finite element methods of Dirichlet boundary-value problems of steady-state fractional diffusion equations under the assumption that the true solutions have desired regularity and that the solution to the dual problem has full regularity for each right-hand side. We show that the solution to the homogeneous Dirichlet boundary-value problem of a one-dimensional steady-state fractional diffusion equation of constant coefficient and source term is not necessarily in the Sobolev space $H^1$. This fact has the following implications: (i) Up to now, there are no verifiable conditions on the coefficients and source terms of fractional diffusion equations in the literature to ensure the high regularity of the true solutions, which are in turn needed to guarantee the high-order convergence rates of their numerical approximations. (ii) Any Nitsche-lifting based proof of optimal-order $L^2$ error estimates of finite element methods in the literature is invalid. We present numerical results to show that high-order finite element methods for a steady-state fractional diffusion equation with smooth data and source term fail to achieve high-order convergence rates. We present a preliminary development of an indirect finite element method, which reduces the solution of fractional diffusion equations to that of second-order diffusion equations postprocessed by a fractional differentiation. We prove that the corresponding high-order methods achieve high-order convergence rates even though the true solutions are not smooth, provided that the coefficient and source term of the problem have desired regularities. Numerical experiments are presented to substantiate the theoretical estimates.