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

TL;DR

This paper introduces a direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger equations, demonstrating stability, high-order accuracy, and ease of implementation through theoretical proofs and numerical experiments.

Contribution

The paper develops a novel DDG finite element method specifically for fractional PDEs with fractional Laplacian, including stability and convergence analysis.

Findings

Proves stability and optimal convergence order of O(h^{N+1})

Demonstrates high-order accuracy through numerical experiments

Simplifies formulation and implementation of fractional PDE solutions

Abstract

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.