This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by \emph{a posteriori} error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical.

翻译：本文提出并分析了一种用于求解高异质介质中单相非线性可压缩流的约束能量最小化广义多尺度有限元方法（CEM-GMsFEM）。CEM-GMsFEM的构建依赖于两个关键步骤：首先，通过求解局部谱问题构建辅助空间，捕获对应小特征值的基函数；其次，利用该辅助空间在超采样区域上求解局部能量最小化问题获得基函数。这些基函数在相应局部超采样区域外呈指数衰减。本文给出了所提方法的收敛性证明，并表明该收敛性仅依赖于粗网格尺寸，而与介质的异质性无关。为提升计算效率，还开发了一种基于后验误差估计器的在线富集策略。通过三维算例的数值实验验证了理论结果，展示了该方法的性能，并获得了高效且精确的数值结果。