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

TL;DR

This paper considers a time-dependent discrete network model with highly varying connectivity, and proposes the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method.

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.

Figures (18)

• Figure 1: Nodes and connections (edges)
• Figure 2: Illustration of the 2D and 3D networks and corresponding graph Laplacian.
• Figure 3: Illustration of two and three-dimensional fine-scale networks embedded into the cube
• Figure 4: Illustration of coarse grid $\mathcal{T}_H$ and local domain $\omega_i$ with sub-network $G^{\omega_i}$
• Figure 5: Schematic flow chart of the multiscale scheme.

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• Theorem 1