A Hybridizable Discontinuous Galerkin Method for Magnetic Advection-Diffusion Problems

We propose and analyze a hybridizable discontinuous Galerkin (HDG) method for solving a mixed magnetic advection-diffusion problem within a more general Friedrichs system framework. With carefully constructed numerical traces, we introduce two distinct stabilization parameters: tau t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _t$$\end{document} for the tangential trace and tau n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _n$$\end{document} for the normal trace. These parameters are tailored to satisfy different requirements, ensuring the stability and convergence of the method. Furthermore, we incorporate a weight function to facilitate the establishment of stability conditions. We also investigate an elementwise postprocessing technique that proves to be effective for both two-dimensional and three-dimensional problems in terms of broken H(curl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{H}}(\textrm{curl})$$\end{document} semi-norm accuracy improvement. Extensive numerical examples are presented to showcase the performance and effectiveness of the HDG method and the postprocessing techniques.