Generalized multiscale finite element method for language competition modeling II: Online approach.

In this paper, we consider language competition in a heterogeneous medium. The mathematical model considers bilingualism, population growth, and social pressure to use a specific language (diglossia). The transition from one language group to another is determined by probabilities that depend on status, group size, and spatial location. The model considers not only diffusive spread but also convective spread, which models seasonal migrations of speaker populations. We develop multiscale approaches for coarse grid modeling while maintaining reasonable accuracy. We present offline and online approaches based on the Generalized Multiscale Finite Element Method. We consider a model problem in a heterogeneous medium to test these approaches. The results show that both approaches can significantly reduce the discrete problem's size. However, the online approach achieves better accuracy. • We consider a language competition process between two languages in a heterogeneous medium. • The mathematical model considers diffusive-convective spatial spread, bilingualism, logistic growth, and diglossia. • For efficient modeling, we present offline and online multiscale approaches based on the GMsFEM and the Residual-Driven Online GMsFEM, respectively. • Numerical results show that both multiscale approaches have high accuracy, but the online approach provides better results than the offline one. • The online adaptive strategy provides accuracy similar to the uniform one but with fewer degrees of freedom. [ABSTRACT FROM AUTHOR]