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In this paper, we present an upscaling method for problems in perforateddomains with non-homogeneous boundary conditions on perforations. Ourmethodology is based on the recently developed Non-local multicontinuum method(NLMC). The main ingredient of the method is the construction of suitable localbasis functions with the capability of capturing multiscale features andnon-local effects. We will construct multiscale basis functions for the coarseregions and additional multiscale basis functions for perforations, with theaim of handling non-homogeneous boundary conditions on perforations. We startwith describing our method for the Laplace equation, and then extending theframework for the elasticity problem and parabolic equations. The resultingupscaled model has minimal size and the solution has physical meaning on thecoarse grid. We will present numerical results (1) for steady and unsteadyproblems, (2) for Laplace and Elastic operators, and (3) for Neumann and Robinnon-homogeneous boundary conditions on perforations. Numerical results showthat the proposed method can provide good accuracy and provide significantreduction on the degrees of freedom.

Upscaling method for problems in perforated domains with non-homogeneous  boundary conditions on perforations using Non-Local Multi-Continuum method  (NLMC)