Non-local multicontinua upscaling

In this chapter, we design new upscaled models based on CEM-GMsFEM and discuss the relation between multiscale and upscaled models. The degrees of freedom in CEM-GMsFEM or in multiscale methods do not have physical meanings since they are coordinates in the multiscale space. In order to obtain an upscaled model, one needs to design multiscale basis functions such that the degrees of freedom represent the average of the solutions. This allows relating multiscale methods to upscaled models and one does not need to use basis functions in further updated. In this chapter, we will discuss some of these methods, where the basis functions are constructed using some physics-based constraints. The resulting upscaled models share some similarities with traditional multicontinuumMulticontinuum approaches. While choosing different constraints in multiscale methods, one can obtain various upscaled models. This is a key part, which allows connecting multiscale methods and upscaling approaches.