Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

TL;DR

This paper develops an asymptotic expansion for steady-state convection-diffusion solutions in thin, graph-like 3D structures with small diffusion, providing rigorous estimates of approximation accuracy.

Contribution

It introduces a multiscale analysis method to construct and justify asymptotic solutions for convection-diffusion problems in thin graph-like domains with explicit error bounds.

Findings

Asymptotic expansion accurately approximates the solution as   small.

Proved Sobolev and uniform norm estimates for the approximation error.

Method applicable to complex thin network geometries.

Abstract

A steady-state convection-diffusion problem with a small diffusion of order $\mathcal{O}(\varepsilon)$ is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and justified. The asymptotic estimates in the norm of Sobolev space $H^1$ as well as in the uniform norm are proved for the difference between the solution and proposed approximations with a predetermined accuracy with respect to the degree of $\varepsilon$.