  This paper presents a hybrid numerical method to solve efficiently a class of
highly anisotropic elliptic problems. The anisotropy is aligned with one
coordinate-axis and its strength is described by a parameter $\eps \in (0,1]$,
which can largely vary in the study domain. Our hybrid model is based on
asymptotic techniques and couples (spatially) an Asymptotic-Preserving model
with its asymptotic Limit model, the latter being used in regions where the
anisotropy parameter $\eps$ is small. Adequate coupling conditions link the two
models. Aim of this hybrid procedure is to reduce the computational time for
problems where the region of small $\eps$-values extends over a significant
part of the domain, and this due to the reduced complexity of the limit model.
