In this paper, we study the generalized multiscale finite element method (GMsFEM) for single phase compressible flow in highly heterogeneous porous media. We follow the major steps of the GMsFEM to construct permeability dependent offline basis for fast coarse-grid simulation. The offline coarse space is efficiently constructed only once based on the initial permeability field with parallel computing. A rigorous convergence analysis is performed for two types of snapshot spaces. The analysis indicates that the convergence rates of the proposed multiscale method depend on the coarse meshsize and the eigenvalue decay of the local spectral problem. To further increase the accuracy of multiscale method, residual driven online multiscale basis is added to the offline space. The construction of online multiscale basis is based on a carefully design error indicator motivated by the analysis. We find that online basis is particularly important for the singular source. Rich numerical tests on typical 3D highly heterogeneous medias are presented to demonstrate the impressive computational advantages of the proposed multiscale method.
翻译:在本文中,我们研究了在高度多样化的多孔化介质中,单一阶段压缩流的通用多级有限元素法(GMSFEM),我们遵循GMSFEM的主要步骤,为快速粗格网络模拟建立基于离线的渗透性依赖离线基础。离线粗略空间只有一次根据初始渗透性字段和平行计算有效构建。对两种类型的快照空间进行了严格的趋同分析。分析表明,拟议的多级方法的趋同率取决于粗略网格和本地光谱问题的二元值衰减。为了进一步提高多尺度方法的准确性,在离线空间中添加了剩余驱动的在线多尺度基础。在线多尺度基础的构建基于分析所激发的仔细设计误差指标。我们认为,在线基础对于单一源尤其重要。对典型的3D高度多样化媒体进行了丰富的数字测试,以展示拟议的多尺度方法的令人印象深刻的计算优势。