SFB 1313 Publication "Asymptotic expansion for convection-dominated transport in a thin graph-like junction"

New SFB 1313 publication, published in "Analysis and Applications". The work has been developed by researchers involved in Project Areas B, and an internal research project. 

Asymptotic expansion for convection-dominated transport in a thin graph-like junction

Authors

• Taras Mel’nyk (University of Stuttgart, research project I-08) 
• Christian Rohde (University of Stuttgart, research project B-03)

Abstract

We consider a parabolic convection–diffusion problem with small Péclet number in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with small radii  connected through a domain (node). Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces  of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.
< 1 , which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε 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 .