We consider the damped time-harmonic Galbrun's equation, which is used to model stellar oscillations. We introduce a discontinuous Galerkin finite element method (DGFEM) with $H(\operatorname{div})$-elements, which is nonconform with respect to the convection operator. We report a convergence analysis, which is based on the frameworks of discrete approximation schemes and T-compatibility. A novelty is that we show how to interprete a DGFEM as a discrete approximation scheme and this approach enables us to apply compact perturbation arguments in a DG-setting, and to circumvent any extra regularity assumptions on the solution. The advantage of the proposed $H(\operatorname{div})$-DGFEM compared to $H^1$-conforming methods is that we do not require a minimal polynomial order or any special assumptions on the mesh structure. The considered DGFEM is constructed without a stabilization term, which considerably improves the assumption on the smallness of the Mach number compared to other DG methods and $H^1$-conforming methods, and the obtained bound is fairly explicit. In addition, the method is robust with respect to the drastic changes of magnitude of the density and sound speed, which occur in stars. The convergence of the method is obtained without additional regularity assumptions on the solution, and for smooth solutions and parameters convergence rates are derived.

翻译：我们考虑用于模拟恒星振荡的阻尼时间谐波Galbrun方程。我们引入一种采用$H(\operatorname{div})$元的间断Galerkin有限元方法(DGFEM)，该方法在对流算子意义下是非协调的。基于离散逼近方案和T-相容性框架，我们报告了收敛性分析。一个创新点在于我们展示了如何将DGFEM解释为离散逼近方案，这种方法使我们能够在DG框架下应用紧摄动论证，并避免对解施加任何额外的正则性假设。与$H^1$协调方法相比，所提出的$H(\operatorname{div})$-DGFEM的优势在于我们不需要最低多项式阶数或对网格结构进行任何特殊假设。所考虑的DGFEM在无稳定项的情况下构建，与其他DG方法和$H^1$协调方法相比，这显著改善了对马赫数小量性的假设，且得到的界相当显式。此外，该方法对恒星中出现的密度和声速量级的剧烈变化具有鲁棒性。该方法在无需对解施加额外正则性假设的情况下实现收敛，并且对于光滑解和参数，推导出了收敛阶。