Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions

Article type: Research Article

Authors: Mel’nyk, Taras^{; *} | Rohde, Christian

Affiliations: Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Correspondence: [*] Corresponding author. E-mail: Taras.Melnyk@mathematik.uni-stuttgart.de.

Abstract: This article completes the study of the influence of the intensity parameter α in the boundary condition ε∂νεuε−uεVε→·νε=εαφε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O(ε−1) is considered. The novelty of this article is the case of α<1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φε (the cases α=1 and α>1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution uε as ε→0, i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.

Keywords: Asymptotic expansion, convection-diffusion problem, boundary interactions, thin graph-like junction, hyperbolic limit model

DOI: 10.3233/ASY-231876

Journal: Asymptotic Analysis, vol. 137, no. 1-2, pp. 27-52, 2024