An adaptive spectral graph wavelet method for PDEs on networks

Abstract

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.