Integro-differential equations, analyzed in this work, comprise an important class of models of continuum media with nonlocal interactions. Examples include peridynamics, population and opinion dynamics, the spread of disease models, and nonlocal diffusion, to name a few. They also arise naturally as a continuum limit of interacting dynamical systems on networks. Many real-world networks, including neuronal, epidemiological, and information networks, exhibit self-similarity, which translates into self-similarity of the spatial domain of the continuum limit. For a class of evolution equations with nonlocal interactions on self-similar domains, we construct a discontinuous Galerkin method and develop a framework for studying its convergence. Specifically, for the model at hand, we identify a natural scale of function spaces, which respects self-similarity of the spatial domain, and estimate the rate of convergence under minimal assumptions on the regularity of the interaction kernel. The analytical results are illustrated by numerical experiments on a model problem.

翻译：本工作分析的非局部积分-微分方程是连续介质中具有非局部相互作用的重要模型类别的组成部分，其范例包括近场动力学、群体与观点动力学、疾病传播模型以及非局部扩散等。这类方程也自然产生于网络上相互作用动力系统的连续极限。许多现实世界中的网络（包括神经元网络、流行病学网络和信息网络）均表现出自相似性，这种性质映射到连续极限空间域的自相似性中。针对自相似域上一类具有非局部相互作用的演化方程，我们构建了一种间断伽辽金方法，并建立了研究其收敛性的框架。具体而言，针对所研究的模型，我们确定了一组尊重空间域自相似性的自然函数空间尺度，并在交互核正则性的最小假设下估计了收敛速率。通过模型问题的数值实验验证了理论分析结果。