A direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger-type equations

Abstract

A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schrödinger-type equations with a fractional Laplacian operator of order $\alpha$ $(1 < \alpha < 2)$ . The fractional operator of order $\alpha$ is expressed as a composite of second-order derivative and a fractional integral of order $2-\alpha$ . These problems have been expressed as a system of parabolic equations and low-order integral equation. This allows us to apply the DDG method, which is based on the direct weak formulation for solutions of fractional convection-diffusion and Schrödinger-type equations in each computational cell, letting cells communicate via the numerical flux $(\partial_{x}u)^{\ast}$ only. Moreover, we prove stability and optimal order of convergence $ O(h^{N+1})$ for the general fractional convection-diffusion and Schrödinger problems, where h, N are the space step size and polynomial degree. The DDG method has the advantage of easier formulation and implementation as well as the high-order accuracy. Finally, numerical experiments confirm the theoretical results of the method.