A direct discontinuous Galerkin method for fractional convection-diffusion and Schr\"{o}dinger type equations

A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schr\"{o}dinger 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 equation 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 Schr\"{o}dinger type equations in each computational cell, letting cells communicate via the numerical flux $(\partial_{x}u)^{*}$ only. Moreover, we prove stability and optimal order of convergence $O(h^{N+1})$ for the general fractional convection-diffusion and Schr\"{o}dinger 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.