Discontinuous Galerkin method for fractional convection-diffusion equations

Abstract

We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order $α(1<α<2)$ defined through the fractional Laplacian. The fractional operator of order $α$ is expressed as a composite of first order derivatives and fractional integrals of order $2-α$, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stability and optimal order of convergence O($h^{k+1}$) for subdiffusion, and an order of convergence of ${\cal O}(h^{k+1/2})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.