Generalized Multiscale Finite Element Method for discrete network (graph) models

Abstract: In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.