A direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger type equations
This work provides a novel numerical method for solving fractional partial differential equations, which are important in modeling anomalous diffusion and quantum mechanics, but the contribution is incremental as it extends existing DDG techniques to a new class of problems.
The authors developed a direct discontinuous Galerkin (DDG) method for fractional convection-diffusion and Schrödinger equations with a fractional Laplacian, achieving optimal convergence order O(h^{N+1}) and stability, validated by numerical experiments.
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 $α$ $(1<α<2)$. The fractional operator of order $α$ is expressed as a composite of second order derivative and a fractional integral of order $2-α$. 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 Schrödinger 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 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.