Generalized multiscale finite element method for language competition modeling I: Offline approach

Abstract

This paper develops a multiscale solver for the problem of two languages competing in a heterogeneous medium. The mathematical model contains terms for language group switching, diglossia, population growth, and diffusive-convective spatial spread. The fine grid approximation is based on the finite element method. We use the finite difference method with an implicit scheme for time approximation. We use the Picard method to linearize the nonlinear terms. We use the Generalized Multiscale Finite Element Method for approximation on a coarse grid. Numerical results are presented for a model two-language interaction problem. The results show that the proposed multiscale approach can significantly reduce the computational cost of the problem while maintaining high accuracy.