A Generalized Multiscale Finite Element Method (GMsFEM) for perforated domain flows with Robin boundary conditions

Abstract

In this work, we develop a multiscale method for elliptic problems in perforated media with Robin boundary conditions. Perforated media are encountered in many applications, e.g., in material science applications, where the material properties are prescribed outside perforation. Such a perforated structure can significantly affect the solution. For the numerical solution of the equation, it is necessary to construct a computational grid with elements resolving the perforations. The computational grid calculated in this way is fine-scale grid and this can lead to a large number of unknowns. For the numerical solution of our problem, we construct an approximation of the equation on a coarse grid using the Generalized Multiscale Finite Element Method (GMsFEM). The main idea of the GMsFEM is to construct multiscale basis functions on a coarse grid, which can reduce the computational cost. The multiscale basis function construction requires snapshots and local spectral problems, which are developed in the paper. In the paper, we study Robin boundary conditions, which arise in many applications. Previous works (Chung et al., 2016) considered Dirichlet and Neumann boundary conditions. The construction of multiscale basis functions differs from those in Chung et al. 2016 [1] , [2] and as we need to impose non-homogeneous Robin boundary conditions. We present several numerical examples. In these examples, perforated domains with many inclusions are considered. Our numerical results show a good agreement between the coarse and the fine-grid simulations.