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

TL;DR

This paper introduces a generalized multiscale finite element method for time-dependent discrete network models with highly variable connectivity, providing accurate coarse-scale approximations and convergence analysis.

Contribution

It develops a novel multiscale finite element approach tailored for complex network models, including spectral problem-based coarse approximation and stability analysis.

Findings

Effective coarse-scale approximation for heterogeneous networks

Proven convergence and stability of the method

Numerical validation on structured and random networks

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.