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A nonuniform Neumann boundary-value problem is considered for the Poissonequation in a thin $3D$ aneurysm-type domain that consists of thin curvilinearcylinders that are joined through an aneurysm of diameter$\mathcal{O}(\varepsilon).$  A rigorous procedure is developed to construct the complete asymptoticexpansion for the solution as the parameter $\varepsilon \to 0.$  The asymptotic expansion consists of a regular part that is located inside ofeach cylinder, a boundary-layer part near the base of each cylinder, and aninner part discovered in a neighborhood of the aneurysm. Terms of the innerpart of the asymptotics are special solutions of boundary-value problems in anunbounded domain with different outlets at infinity. It turns out that theyhave polynomial growth at infinity. By matching these parts, we derive thelimit problem $(\varepsilon =0)$ in the corresponding graph and a recurrenceprocedure to determine all terms of the asymptotic expansion.  Energetic and uniform pointwise estimates are proved. These estimates allowus to observe the impact of the aneurysm.

Asymptotic approximations of the solution to a boundary-value problem in a thin aneurysm-type domain