Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations
This work provides a rigorous adaptive framework for fractional PDEs, which is important for computational scientists working on nonlocal models.
The paper develops adaptive discontinuous Galerkin methods for tempered fractional (convection) diffusion equations, providing stability and convergence analyses with posteriori error estimates. Numerical experiments confirm the effectiveness of the adaptive schemes.
This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are respectively verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.