Volume 36, Issue 3
Effective Boundary Conditions for the Heat Equation with Interior Inclusion

Commun. Math. Res., 36 (2020), pp. 272-295.

Published online: 2020-07

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• Abstract

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 "effective boundary conditions" (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.

• Keywords

Heat equation, effective boundary conditions, weak solution, a priori estimates, asymptotic behavior.

35K05, 35B40, 35B45, 74K15

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@Article{CMR-36-272, author = {Huicong and Li and and 8392 and and Huicong Li and Xuefeng and Wang and and 8393 and and Xuefeng Wang}, title = {Effective Boundary Conditions for the Heat Equation with Interior Inclusion}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {3}, pages = {272--295}, abstract = {

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 "effective boundary conditions" (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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0012}, url = {http://global-sci.org/intro/article_detail/cmr/17849.html} }
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 -

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 "effective boundary conditions" (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.

Huicong Li & Xuefeng Wang. (2020). Effective Boundary Conditions for the Heat Equation with Interior Inclusion. Communications in Mathematical Research . 36 (3). 272-295. doi:10.4208/cmr.2020-0012
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