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

摘要

A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schrodinger-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 Schrodinger-type equations in each computational cell, letting cells communicate via the numerical flux (partial derivative(x)u)* only. Moreover, we prove stability and optimal order of convergence O(h(N+1) ) for the general fractional convection-diffusion and Schrodinger 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.