Asymptotic modeling of a~problem with contrasting stiffness

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 38, Tome 369 (2009), pp. 164-201

Résumé

An asymptotic model is found of the Neumann problem for second-order differential equation with piecewise constant coefficients in the composite domain $\Omega\cup\omega$ which are small of order $O(\varepsilon)$ in the subdomain $\omega$. Namely a domain $\Omega(\varepsilon)$ with a singular perturbed boundary is constructed whose solution gives a two-term asymptotic, i.e., of the increased accuracy $O(\varepsilon^2)$, approximation solution for the restriction on $\Omega$ of the original problem. In contrast to other singularly perturbed problems, in the case of contrasting stiffness modeling requires for constructing the contour $\partial\Omega(\varepsilon)$ with ledges, i.e., boundary fragments with curvature $O(\varepsilon^{-1})$. Bibl. – 33 titles.