Homogenization of the Robin problem in a thick multilevel junction

Abstract

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε that is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness of order ε = O(N −1 ). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are ε-periodically alternated. We investigate the asymptotic behavior of the solution as ε → 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, a convergence theorem is proved.

Homogenization of a Boundary-Value Problem with Varying type of Boundary Conditions in a Thick Two-Level Junction

“…In [32], with the help of special extension operators, a convergence theorem was proved for a solution of the Poisson equation in a plane two-level junction with homogeneous Robin boundary conditions at the boundaries of thin rods. In [33], with the Taras Shevchenko National University of Kyiv, Kyiv.…”

“…This fact was noted in [38] in the course of the homogenization of elliptic equations that describe processes in strongly inhomogeneous thin perforated domains with rapidly varying thickness. If Dirichlet conditions are replaced by Neumann conditions or Fourier conditions, then, as shown in [32], the boundary-value problem does not decompose and consists, in fact, of three boundary-value problems joined together into a single problem by certain conjugation conditions in the junction zone.…”

“…However, for two-level thick junctions, it is necessary to construct two special operators of extension by zero to different domains. In the case where the thin rods of a two-level thick junction are of variable thickness, it is necessary to construct special extension operators (see [32]). …”

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

“…It has been shown that, due to the homogeneous Dirichlet boundary conditions on the vertical sides of the rods from the first level, the original boundary-value problem (1) decomposes in the limit into the two independent problems (13) and (14). Due to the inhomogeneous Neumann boundary conditions on the vertical sides of the rods from the second level, we have obtained the corresponding term on the right-hand side of the ordinary differential equation of problem (14).…”

“…These thick junctions have a more complex structure, and, for this reason, the asymptotic investigation has its own specific features and new qualitative results (see [13][14][15][16][17][18]). Here we construct an asymptotic approximation for a solution of a mixed boundary-value problem in a thick two-level junction and investigate the influence of the varying type of boundary conditions on the asymptotic behavior.…”