波数显式边界下时谐高频弹性动力学的多尺度亚网格修正方法
Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wavenumber Explicit Bounds
摘要：
弹性动力学方程在高频下的模拟受到众所周知的污染效应的影响。我们提出了一个彼得罗夫-Galerkin多尺度子网格校正方法，保持无污染的自然分辨率和过采样制度。这是通过生成对粗网格空间的校正来实现的，粗网格空间具有由与$\log(k)$相关的过采样长度确定的支持，$k$是波数。这种方法的关键是多项式在$k$边界的稳定常数和相关的inf-sup常数。为此，我们建立多项式的$k$边界的弹性动力学稳定常数在一般Lipschitz域辐射边界条件在$\mathbb{R}^3$。以前的方法依赖于变分技术，Rellich身份，和几何约束。在弹性动力学的背景下，这些遭受需要假设一个科恩的边界上的不平等。在这项工作中的方法是基于边界积分算子和格林函数的导数依赖于$k$的估计，不需要这个额外的假设。我们还实现了二维和三维的数值例子，以显示该方法消除了污染的自然分辨率和过采样制度，以及执行良好的标准拉格朗日有限元相比。
英文摘要：
The simulation of the elastodynamics equations at high-frequency suffers from the well known pollution effect. We present a Petrov--Galerkin multiscale sub-grid correction method that remains pollution free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the $\log(k)$, $k$ being the wavenumber. Key to this method are polynomial-in-$k$ bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-$k$ bounds for the elastodynamics stability constants in general Lipschitz domains with radiation boundary conditions in $\mathbb{R}^3$. Previous methods relied on variational techniques, Rellich identities, and geometric constraints. In the context of elastodynamics, these suffer from the need to hypothesize a Korn's inequality on the boundary. The methods in this work are based on boundary integral operators and estimation of Green's function's derivatives dependence on $k$ and do not require this extra hypothesis. We also implemented numerical examples in two and three dimensions to show the method eliminates pollution in the natural resolution and oversampling regimes, as well as performs well when compared to standard Lagrange finite elements.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
