# Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations

## TL;DR

This paper develops and analyzes adaptive discontinuous Galerkin methods for solving tempered fractional convection-diffusion equations, demonstrating their stability, convergence, and effectiveness through theoretical analysis and numerical experiments.

## Contribution

It introduces a novel adaptive DG framework with a posteriori error estimates for tempered fractional PDEs, applicable to general fractional operators.

## Key findings

- Proved stability and convergence of the proposed DG schemes.
- Designed effective local error indicators for adaptivity.
- Validated the methods through extensive numerical experiments.

## Abstract

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.