We present a stable discontinuous Galerkin (DG) method with a perfectly matched layer (PML) for three and two space dimensional linear elastodynamics, in velocity-stress formulation, subject to well-posed linear boundary conditions. First, we consider the elastodynamics equation, in a cuboidal domain, and derive an unsplit PML truncating the domain using complex coordinate stretching. Leveraging the hyperbolic structure of the underlying system, we construct continuous energy estimates, in the time domain for the elastic wave equation, and in the Laplace space for a sequence of PML model problems, with variations in one, two and three space dimensions, respectively. They correspond to PMLs normal to boundary faces, along edges and in corners. Second, we develop a DG numerical method for the linear elastodynamics equation using physically motivated numerical flux and penalty parameters, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Third, to ensure numerical stability of the discretization when PML damping is present, it is necessary to extend the numerical DG fluxes, and the numerical inter-element and boundary procedures, to the PML auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. By combining the DG spatial approximation with the high order ADER time stepping scheme and the accuracy of the PML we obtain an arbitrarily accurate wave propagation solver in the time domain. Numerical experiments are presented in two and three space dimensions corroborating the theoretical results.

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一阶形式的弹性动力学完美匹配层的稳定不连续伽辽金方法

我们提出了一种稳定的不连续伽辽金 (DG) 方法，具有完美匹配层 (PML)，适用于三维和二维线性弹性动力学，在速度-应力公式中，受适定线性边界条件的约束。首先，我们考虑立方体域中的弹性动力学方程，并推导出使用复坐标拉伸截断域的未分裂 PML。利用基础系统的双曲线结构，我们在弹性波动方程的时域和拉普拉斯空间中为一系列 PML 模型问题构建连续的能量估计，分别具有一个、两个和三个空间维度的变化. 它们对应于垂直于边界面、沿边和角的 PML。第二，我们使用物理激励的数值通量和惩罚参数开发了线性弹性动力学方程的 DG 数值方法，这些方法与所有适定的内部和外部边界条件兼容。当 PML 阻尼消失时，通过构造，我们对惩罚参数的选择会产生逆风方案和类似于连续能量估计的离散能量估计。第三，当存在 PML 阻尼时，为了确保离散化的数值稳定性，有必要将数值 DG 通量以及数值单元间和边界程序扩展到 PML 辅助微分方程。这对于导出类似于连续能量估计的离散能量估计是至关重要的。通过将 DG 空间近似与高阶 ADER 时间步进方案和 PML 的精度相结合，我们在时域中获得了任意精确的波传播求解器。数值实验在两个和三个空间维度上进行，证实了理论结果。