Multiscale methods for solving wave equations on spatial networks

TL;DR

This paper introduces a multiscale finite element method for wave equations on spatial networks, combining localized orthogonal decomposition with energy conserving time schemes, achieving optimal error bounds.

Contribution

It develops a novel multiscale approach using LOD on networks for wave problems, integrating spatial and temporal discretizations with proven error bounds.

Findings

Optimal order error bounds derived

Numerical experiments confirm theoretical results

Method conserves energy during wave propagation

Abstract

We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using the localized orthogonal decomposition (LOD) methodology. The spatial discretization is then combined with an energy conserving temporal scheme to form the proposed method. Under the assumption of well-prepared initial data, we derive an a priori error bound of optimal order with respect to the space and time discretization. In the analysis, we combine the theory derived for stationary elliptic problems on spatial networks with classical finite element results for hyperbolic problems. Finally, we present numerical experiments that confirm our theoretical findings.