A mathematical model and numerical solution of interface problems for steady state heat conduction

Abstract

We study interface (or transmission) problems arising in the steady state heat conduction for layered medium. These problems are related to the elliptic equation of the formAu:=−∇(k(x)∇u(x))=F(x),x∈Ω⊂ℝ2, with discontinuous coefficientk=k(x). We analyse two types of jump (or contact) conditions across the interfacesΓδ−=Ω1∩ΩδandΓδ+=Ωδ∩Ω2of the layered mediumΩ:=Ω1∪Ωδ∪Ω2. An asymptotic analysis of the interface problem is derived for the case when the thickness (2δ>0) of the layer (isolation)Ωδtends to zero. For each case, the local truncation errors of the used conservative finite difference schemeareestimated on the nonuniform grid. A fast direct solver has been applied for the interface problems with piecewise constant but discontinuous coefficientk=k(x). The presented numerical results illustrate high accuracy and show applicability of the given approach.