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

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 $\varepsilon$.