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

Abstract

Problems involving materials with thin layers arise in various application fields. We present here the use of asymptotic expansions for linear elliptic problems to derive and justify so-called ap-proximate or effective boundary conditions. We first recall the known results of the literature, and then discuss the optimality of the error estimates in the smooth case. For non-smooth geometries, the results of [18, 57] are commented and adapted to a model problem, and two improvements of the approximate model are proposed to increase its numerical performance.

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

“…This construction is based on the variational formulation of the problem and leads to suboptimal error estimates, which can be easily improved; see Vial 47 and Auvray and Vial. 48 As already mentioned, in the situation of Figure 1, the limit problem reads…”

“…In Bendali and Lemrabet, a complete asymptotic expansion is built in the case where the thin coating lies on the whole boundary of a smooth object. This construction is based on the variational formulation of the problem and leads to suboptimal error estimates, which can be easily improved; see Vial and Auvray and Vial …”

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

“…It is well known that there are two main approaches for the study of problems involving thin layers. We can consider directly the thin layers problems and use adapted numerical methods (see, for example [2,10,15]), or we incorporate the thin layer effect through an approximate boundary condition in an approximate way, see for example [5,8,13] and the references therein. The second approach will be illustrated in this work by the study of a three-dimensional thermoelasticity coupled system for an elastic body covered on one of its faces by a thin layer or a thin shell of thickness ε.…”

Effective interface conditions for continuum mechanical models describing the invasion of multiple cell populations through thin membranes

“…films or membranes) arise in the mathematical modelling of various chemical, physical and biological systems [1,2,3,5,9,10,11,12,14,16,17,20,22,23,25,26,27,28]. Due to the analytical and numerical challenges posed by the presence of such layers [8], it is often convenient to approximate the original problem by an equivalent transmission problem whereby each thin layer is replaced by an effective interface. The equivalent problem is then closed by imposing suitable transmission conditions on the effective interfaces.…”