A Multiscale Method for Porous Microstructures

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.