An embedded SDG method for the convection-diffusion equation

TL;DR

This paper introduces an embedded SDG method for the convection-diffusion equation that combines advantages of SDG and EDG, offering stability, conservation, and efficiency, especially for convection-dominated problems.

Contribution

The paper proposes a novel embedded SDG method that ensures conservation, stability, and high efficiency without complex stabilization, improving upon existing DG methods.

Findings

Achieves optimal convergence in potential

Provides $L^2$ stability with skew-symmetric discretization

Demonstrates good performance in numerical tests

Abstract

In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method.