TY - JOUR
T1 - Effective Boundary Conditions for the Heat Equation with Interior Inclusion
AU - Li , Huicong
AU - Wang , Xuefeng
JO - Communications in Mathematical Research 
VL - 3
SP - 272
EP - 295
PY - 2020
DA - 2020/07
SN - 36
DO - http://doi.org/10.4208/cmr.2020-0012
UR - https://global-sci.org/intro/article_detail/cmr/17849.html
KW - Heat equation, effective boundary conditions, weak solution, a priori estimates, asymptotic behavior.
AB - <p style="text-align: justify;">Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically different
from that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of the
thin layer as a thickless surface, on which we impose &quot;effective boundary conditions&quot; (EBCs). These boundary conditions not only ease the computational
burden, but also reveal the effect of the inclusion. In this paper, by considering
the asymptotic behavior of the heat equation with interior inclusion subject to
Dirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which include
a Poisson equation on the curve, and a non-local one. It turns out that the
EBCs depend on the magnitude of the thermal conductivity in the thin layer,
compared to the reciprocal of its thickness.</p>