标题： 阻尼时谐 Galbrun 方程非协调$H(\operatorname{div})$有限元的收敛性分析 显示英文标题

标题： Convergence analysis of nonconform $H(\operatorname{div})$-finite elements for the damped time-harmonic Galbrun's equation

摘要： 我们考虑阻尼时谐Galbrun方程，该方程用于模拟恒星振荡。 我们引入了一种使用$H(\operatorname{div})$元的不连续伽辽金有限元方法（DGFEM），该方法在对流算子上是非协调的。 我们报告了一个收敛性分析，该分析基于离散逼近方案和T相容性的框架。 新颖之处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设置中应用紧致扰动论证，并避免对解的额外正则性假设。 所提出的$H(\operatorname{div})$-DGFEM相比$H^1$协调方法的优势在于我们不需要最小多项式阶数或对网格结构的特殊假设。 所考虑的DGFEM在构造时没有稳定项，这相较于其他DG方法和$H^1$协调方法显著改善了马赫数小的假设，且得到的界限相当明确。 此外，该方法对于恒星中密度和声速大小的剧烈变化具有鲁棒性。 该方法的收敛性是在不增加对解的正则性假设的情况下获得的，并且对于光滑解和参数，得出了收敛速率。 显示英文摘要

摘要： 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.