Hybrid discontinuous Galerkin discretizations for the damped time-harmonic Galbrun’s equation

In this article, we study the damped time-harmonic Galbrun’s equation which models solar and stellar oscillations. We introduce and analyze hybrid discontinuous Galerkin discretizations (HDG) that are stable and optimally convergent for all polynomial degrees greater than or equal to one. The proposed methods are robust with respect to the drastic changes in the magnitude of the coefficients that naturally occur in stars. Out analysis is based on the concept of discrete approximation schemes and weak $T$-compatibility, which exploits the weakly $T$-coercive structure of the equation. Compared to the $H^1$-conforming discretization of [Halla, Lehrenfeld, Stocker, 2022], our method offers improved stability and robustness. Furthermore, it significantly reduces the computational costs compared to the $H(\text{div})$-conforming DG discretization of [Halla, 2023], which has similar stability properties. These advantages make the proposed HDG methods well-suited for astrophysical simulations.