  In this paper, we present an upscaling method for problems in perforated
domains with non-homogeneous boundary conditions on perforations. Our
methodology is based on the recently developed Non-local multicontinuum method
(NLMC). The main ingredient of the method is the construction of suitable local
basis functions with the capability of capturing multiscale features and
non-local effects. We will construct multiscale basis functions for the coarse
regions and additional multiscale basis functions for perforations, with the
aim of handling non-homogeneous boundary conditions on perforations. We start
with describing our method for the Laplace equation, and then extending the
framework for the elasticity problem and parabolic equations. The resulting
upscaled model has minimal size and the solution has physical meaning on the
coarse grid. We will present numerical results (1) for steady and unsteady
problems, (2) for Laplace and Elastic operators, and (3) for Neumann and Robin
non-homogeneous boundary conditions on perforations. Numerical results show
that the proposed method can provide good accuracy and provide significant
reduction on the degrees of freedom.
