A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations

Abstract

Fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior. Recently, spectral Galerkin methods were developed for space-fractional diffusion equations aiming at achieving exponential convergence. An optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity. An optimal order error estimate in the L2 norm was proved via the well known Nitsche lifting technique under the assumption that the true solution to the corresponding boundary-value problem of the fractional diffusion equation has the required regularity for each right-hand side.In this paper we show that the true solution to the Dirichlet boundary-value problem of a conservative fractional diffusion equation of order 2−β with 0<β<1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional Sobolev space H3/2−β in general, but is still in the Besov space B∞3/2−β(L2). Hence, the provable convergence rate of a spectral Galerkin method in the L2 norm is at most of the order O(N−(3/2−β)), where N is the degree of the polynomial space in the numerical method. Numerical experiments show that the spectral Galerkin method exhibits a subquadratic convergence in the L2 norm for any 0<β<1.We develop a high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of one-sided variable-coefficient conservative fractional diffusion equations. The method has a proved high-order convergence rate of arbitrary order (i) without requiring the smoothness of the true solution u to the given boundary-value problem, but only assuming that the diffusivity coefficient and the right-hand source term have the desired regularity; (ii) for a variable diffusivity coefficient; and (iii) for an inhomogeneous Dirichlet boundary condition. Numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right-hand source term have the desired regularity.