Galerkin method for nonlocal diffusion equations on self-similar domains

TL;DR

This paper develops a discontinuous Galerkin method for nonlocal diffusion equations on self-similar domains, providing convergence analysis and numerical validation for models with nonlocal interactions.

Contribution

It introduces a novel Galerkin framework tailored to self-similar domains and establishes convergence results with minimal kernel regularity assumptions.

Findings

Convergence rate estimates for the proposed method.

Validation through numerical experiments on a model problem.

Framework respects self-similarity in function space analysis.

Abstract

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.