Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.19765

On some coupled local and nonlocal diffusion models

Juan Pablo Borthagaray·Patrick Ciarlet Jr
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted $H^1$-semi norm to model the local part, while two different weighted $H^s$-semi norms, with $s \in (0,1)$, are used to model the nonlocal part. The corresponding strong formulations are derived. In doing so, one needs to develop some technical tools, such as suitable integration by parts formulas for operators with variable diffusivity, and one also needs to study the mapping properties of the Neumann operators that arise. In contrast to problems coupling purely local models, in which one requires transmission conditions on the interface between the subdomains, the presence of a nonlocal operator may give rise to nonlocal fluxes. These nonlocal fluxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specific problem, that we consider most relevant, and study regularity of solutions and finite element discretizations. We provide numerical experiments to illustrate the most salient features of the models.

Explore related subjects

math.NAcs.NAmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Juan Pablo Borthagaray, Patrick Ciarlet Jr. 2025-05-26. On some coupled local and nonlocal diffusion models. https://arxiv.org/abs/2505.19765

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗