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Preprint
Galerkin method for nonlocal diffusion equations on self-similar domains
arXiv.org
12 Dec 2023
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.
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Details
- Title
- Galerkin method for nonlocal diffusion equations on self-similar domains
- Creators
- Georgi S Medvedev
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021863171204721