In this article, we propose an adaptive spectral graph wavelet method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph Laplacian. The beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction of the spectral graph wavelets. Two test functions on different topologies of the network are considered in order to explain the features of the spectral graph wavelet (i.e., behavior of wavelet coefficients and reconstruction error). Subsequently, the method is applied to parabolic problems on networks with different topologies. The numerical results show that the method accurately captures the emergence of the localized patterns at all the scales (including the junction of the network) and the node arrangement is accordingly adapted. The convergence of the method is verified and the efficiency of the method is discussed in terms of CPU time.

在本文中，我们提出了一种自适应光谱图小波方法，用于使用所谓的光谱图小波求解类网络结构上的偏微分方程。频谱图小波的概念基于离散图拉普拉斯算子。该方法的优点在于，将同一算子用于微分算子的近似和频谱图小波的构造。为了说明频谱图小波的特征（即小波系数的行为和重构误差），考虑了网络不同拓扑上的两个测试函数。随后，将该方法应用于具有不同拓扑的网络上的抛物线问题。数值结果表明，该方法可以准确地捕获所有规模（包括网络的交界处）局部模式的出现，并相应地调整了节点的排列方式。验证了该方法的收敛性，并根据CPU时间讨论了该方法的效率。