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Combined asymptotic finite-element modeling of thin layers for scalar elliptic problems. (English) Zbl 1230.80011
Summary: Thin layers with material properties which differ significantly from those of the adjacent media appear in a variety of applications, as in the form of fiber coatings in composite materials. Fully modeling of such thin layers by standard finite element (FE) analysis is often associated with difficult meshing and high computational cost. Asymptotic procedures which model such thin domains by an interface of no thickness on which appropriate interface conditions are devised have been known in the literature for some time. The present paper shows how the first-order asymptotic interface model proposed by Bövik in 1994, and later generalized by Benveniste, can be incorporated in a FE formulation, to yield an accurate and efficient computational scheme for problems involving thin layers. This is done here for linear scalar elliptic problems in two dimensions, prototyped by steady-state heat conduction. Moreover, it is shown that by somewhat modifying the formulation of the Bövik-Benveniste asymptotic model, the proposed formulation is made to preserve the self-adjointness of the original three-phase problem, thus leading to a symmetric FE stiffness matrix. Numerical examples are presented that demonstrate the performance of the method, and show that the proposed scheme is more cost-effective than the full standard FE modeling of the layer.

##### MSC:
 80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer 80A20 Heat and mass transfer, heat flow (MSC2010) 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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