  The concern of this work is the generalization of an Asymptotic Preserving
method for the highly anisotropic elliptic equations presented in [P. Degond,
A. Lozinski, J. Narski, and C. Negulescu. An asymptotic-preserving method for
highly anisotropic elliptic equations based on a micro-macro decomposition. J.
Comput. Phys., 231(7):2724{2740, 2012]. The limitations of the method
introduced there in are omitted by the introduction of a stabilization term in
the Asymptotic Reformulation. Furthermore, anisotropic error indicators and
mesh adaptation algorithms are proposed and tested allowing to reduce
considerably the number of mesh points required to achieve prescribed
precision. Reported meshes have maximum aspect ratio greater than 500.
