A direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger-type equations
Department of Mathematics, Assiut University, 71516, Assiut, Egypt
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Accepted: 29 June 2018
Published online: 9 August 2018
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 . The fractional operator of order is expressed as a composite of second-order derivative and a fractional integral of order . 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 only. Moreover, we prove stability and optimal order of convergence 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.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2018