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

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 form A u : = − ∇ ( k ( x ) ∇ u ( x ) ) = F ( x ) , x ∈ Ω ⊂ ℝ 2 , with discontinuous coefficient k = k ( x ) . We analyse two types of jump (or contact) conditions across the interfaces Γ δ − = Ω 1 ∩ Ω δ and Γ δ + = Ω δ ∩ Ω 2 of the layered medium Ω : = Ω 1 ∪ Ω δ ∪ Ω 2 . An asymptotic analysis of the interface problem is derived for the case when the thickness ( 2 δ > 0 ) of the layer (isolation) Ω δ tends to zero. For each case, the local truncation errors of the used conservative finite difference scheme are estimated on the nonuniform grid. A fast direct solver has been applied for the interface problems with piecewise constant but discontinuous coefficient k = k ( x ) . The presented numerical results illustrate high accuracy and show applicability of the given approach.