The continuum limit of a system of interacting particles on a convergent family of graphs can be described by a nonlocal evolution equation in the limit as the number of particles goes to infinity. Given the continuum limit, the discrete model can be viewed as a Galerkin approximation of the limiting continuous equation. We estimate the speed of convergence of the Galerkin scheme for the model at hand on Euclidean and fractal domains. The latter are relevant when the underlying family of graphs approximates a fractal. Conversely, this paper proposes a Galerkin scheme for a nonlocal diffusion equation on self--similar domains and establishes its convergence rate. Convergence analysis is complemented with numerical integration results for a model problem on Sierpinski Triangle. The rate of convergence of numerical solutions of the model problem fits well the analytical estimate.

翻译：相互作用粒子系统在收敛图族上的连续极限可由非局部演化方程描述，其中粒子数目趋于无穷大。基于该连续极限，离散模型可视为极限连续方程的伽辽金逼近。本文估算了欧几里得域与分形域上该模型的伽辽金格式收敛速度——当底层图族逼近分形时，后者具有重要研究价值。反之，本文提出自相似域上非局部扩散方程的伽辽金格式并建立了其收敛速率。结合谢尔宾斯基三角模型问题的数值积分结果进行收敛性分析，模型问题的数值解收敛速率与理论估计高度吻合。