  The main purpose of the present paper is to study from a numerical analysis
point of view some robust methods designed to cope with stiff (highly
anisotropic) elliptic problems. The so-called asymptotic-preserving schemes
studied in this paper are very efficient in dealing with a wide range of
$\varepsilon$-values, where $0 < \varepsilon \ll 1$ is the stiffness parameter,
responsible for the high anisotropy of the problem. In particular, these
schemes are even able to capture the macroscopic properties of the system, as
$\varepsilon$ tends towards zero, while the discretization parameters remain
fixed. The objective of this work shall be to prove some
$\varepsilon$-independent convergence results for these numerical schemes and
put hence some more rigor in the construction of such AP-methods.
