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.

翻译：本文针对空间网络上的波动传播问题，提出并分析了一种多尺度方法。通过引入粗尺度并利用插值到网络上的有限元空间，我们采用局域化正交分解（LOD）方法论构建了一个离散多尺度空间。随后，将该空间离散化方案与能量守恒的时间格式相结合，形成了所提出的方法。在初始数据充分光滑的假设下，推导了关于空间和时间离散化的最优阶先验误差界。分析过程中，我们将空间网络上稳态椭圆问题的理论与双曲问题经典有限元结果相结合。最后，通过数值实验验证了理论分析结论。