Discontinuous Galerkin methods for magnetic advection-diffusion problems

TL;DR

This paper introduces a new class of primal discontinuous Galerkin methods for magnetic advection-diffusion problems, offering enhanced stability and error estimates through innovative mechanisms and theoretical analysis.

Contribution

It presents a novel framework for DG methods with a new stabilization mechanism and provides rigorous stability and error analysis for magnetic advection-diffusion problems.

Findings

Proposed a new DG scheme with improved stability.

Established optimal error estimates for the methods.

Numerical experiments confirm theoretical results.

Abstract

We devise and analyze a class of the primal discontinuous Galerkin methods for the magnetic advection-diffusion problems based on the weighted-residual approach. In addition to the upwind stabilization, we find a new mechanism under the vector case that provides more flexibility in constructing the schemes. For the more general Friedrichs system, we show the stability and optimal error estimate, which boil down to two core ingredients -- the weight function and the special projection -- that contain information of advection. Numerical experiments are provided to verify the theoretical results.