Constraint Energy Minimizing Generalized Multiscale Finite Element Method for High-Contrast Linear Elasticity Problem

In this paper, we consider the offline and online Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for high-contrast linear elasticity problem. Offline basis construction starts with an auxiliary multiscale space by solving local spectral problems. We select eigenfunctions that correspond to a few small eigenvalues to form the auxiliary space. Using the auxiliary space, we solve a constraint energy minimization problem to construct offline multiscale spaces. The minimization problem is defined in the oversampling domain, which is larger than the target coarse block. To get a good approximation space, the oversampling domain should be large enough. We also propose a relaxed minimization problem to construct multiscale basis functions, which will yield more accurate and robust solution. To take into account the influence of input parameters, such as source terms, we propose the construction of online multiscale basis and an adaptive enrichment algorithm. We provide extensive numerical experiments on 2D and 3D models to show the performance of the proposed method.