  In this article, we derive the asymptotic expansion, up to an arbitrary order
in theory, 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.
