  In this paper we develop a multiscale method to solve problems in complicated
porous microstructures with Neumann boundary conditions. By using a coarse-grid
quasi-interpolation operator to define a fine detail space and local orthogonal
decomposition, we construct multiscale corrections to coarse-grid basis
functions with microstructure. By truncating the corrector functions we are
able to make a computationally efficient scheme. Error results and analysis are
presented. A key component of this analysis is the investigation of the
Poincar\'{e} constants in perforated domains as they may contain
micro-structural information. Using a constructive method originally developed
for weighted Poincar\'{e} inequalities, we are able to obtain estimates on
Poincar\'{e} constants with respect to scale and separation length of the
pores. Finally, two numerical examples are presented to verify our estimates.
