A Review and Implementation of the Generalized Multiscale Method (GMsFEM)

Abstract

The aim of this article is to study and apply three numerical methods for solving elliptic partial differential equations: the finite element method (FEM), the multiscale finite element method (MsFEM), and the generalized multiscale finite element method (GMsFEM), with special emphasis on the latter. To this end, a theoretical review of their variational and matrix foundations is presented, followed by computational implementations for homogeneous, heterogeneous, and high-contrast diffusion problems. The simulations were carried out using FreeFem++ and MATLAB, which made it possible to compare the performance of the methods under different scenarios. The results show that the classical multiscale method has limitations when the problem involves highly heterogeneous media or non-separable scales, whereas the generalized multiscale method provides a better representation of local behavior through the construction of basis functions obtained from local spectral problems. It is concluded that this latter method is a more suitable alternative for the numerical approximation of complex multiscale problems, especially in high-contrast settings, where reducing degrees of freedom is required without significantly compromising approximation quality.