Asymptotic expansion with boundary layer analysis for strongly anisotropic elliptic equations

In this article, we derive the asymptotic expansion, in theory, up to an arbitrary order for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the Neumann boundary condition and the Dirichlet boundary condition on specific parts of the domain boundary, respectively. The ill-posedness arising from the Neumann boundary condition in the strongly anisotropic diffusion limit is handled by the decomposition of the solution into a mean part and a fluctuation part. The boundary layer analysis due to the Dirichlet boundary condition is conducted for each order in the expansion for the fluctuation part. Our results suggest that the leading order is the combination of the mean part and the composite approximation of the fluctuation part for the general Dirichlet boundary condition. We also apply this method to derive the results for the heterogeneous diffusion problems.

Keywords

strongly anisotropic elliptic equation, matched asymptotic expansion, boundary layer analysis

2010 Mathematics Subject Classification

35B25, 35C20, 35J25

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The research of Xiang Zhou. was supported by the grants from the Research Grants Council of the Hong Kong Special Administrative Region, P.R. China (Project No. CityU 11304314, 11304715 and 11337216).

The research of Ling Lin was partially supported by the research start-up grants from the 100 Top Talents Program of Sun Yat-sen University (No. 34000-18831102).

Received 10 December 2016

Received revised 21 January 2018

Accepted 21 January 2018

Published 30 August 2018