Asymptotic expansion for the solution to a boundary-value problem in a thin cascade domain with a local joint

Abstract

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain $\Omega_\varepsilon$ coinciding with two thin rectangles connected through a joint of diameter ${\cal O}(\varepsilon)$. A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter $\varepsilon \to 0.$ Energetic and uniform pointwise estimates for the difference between the solution of the starting problem $(\varepsilon >0)$ and the solution of the corresponding limit problem $(\varepsilon =0)$ are proved, from which the influence of the geometric irregularity of the joint is observed.