Abstract: Many real-world applications occur in heterogeneous media, for example, subsurface flow and processes in composite materials. For an accurate numerical solution, we should use a very fine grid that resolves all small-scale heterogeneities. To reduce the size of the system, the upscaling technique or multiscale method is used.

In this work, we present the construction of the reduced-order model using the Generalized Multiscale Finite Element Method (GMsFEM) and the NonLocal MultiContinua upscaling technique (NLMC). These methods involve two basic steps: (1) the construction of multiscale basic functions that take into account small-scale heterogeneities in the local domains and (2) the construction of the coarse-scale approximation on a multiscale space. We present numerical results for several applied problems with different types of heterogeneities.