In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problems. Then a Petrov-Galerkin projection and systematization onto the coarse grid is applied. As a result, an explicit and energy conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chose the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.
翻译:在本文中,我们为高度多样化的媒体的弹性波方程式制定了一个本地多尺度的减少模型战略,其实现方式是用一个粗糙的网格和多尺度的功能来解决问题。我们使用内部处罚不连续的Galerkin(IPDG)来将包含重要多元媒体信息的多尺度基函数对齐。高效的多尺度功能的构建始于提取经仔细界定的光谱问题的主要模式,以代表重要的媒体特征,随后解决限制性能最小化的问题。随后,在粗粗格网格上应用了Petrov-Galerkin的投影和系统化。因此,为快速在线模拟取得了一个明确和节能计划。该方法显示粗略和光谱的趋同,只要适当选择了过度抽样大小。我们严格分析拟议方法的稳定性和趋同性。提供了数字结果,以显示多尺度方法的性能并证实理论结果。