Improved impedance conditions for a thin layer problem in a nonsmooth domain

Abstract

In this article, we consider the model problem of the Laplace equation in a domain with a thin layer on a part of its boundary. The singularities appearing where boundary conditions change deteriorate the efficiency of the classical impedance condition used to replace the layer. Modified impedance conditions are proposed, which lead to some improvements in the error estimates.

Asymptotic expansions and effective boundary conditions: a short review for smooth and nonsmooth geometries with thin layers

“…and v 1 can be computed from u 0 reg and Z, via a problem similar to equations (8). It is enough for the following to point out that…”

“…Nevertheless a closer look to the absolute errors show that for the value ρ 0 0.02, even the H 1 -norms are smaller than for ρ 0 = 0 (standard approximate boundary condition). More details on the presented techniques will be found in the forthcoming article [8].…”

Theoretical justification of Ventcel's boundary conditions for a thin layer three-dimensional thermoelasticity problem

“…The analysis of such kind of problems in the two-dimensional case can be found in [4,6,11,12,16,[22][23][24][25]. Let us mention the works [1,3,7,14] in the three-dimensional case.…”