Numerical analysis of an asymptotic-preserving scheme for anisotropic elliptic equations

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 $\epsilon$-values, where $0<\epsilon \ll 1$ 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 $\epsilon$ tends towards zero, while the discretization parameters remain fixed. The objective of this work shall be to prove some $\epsilon$-independent convergence results for these numerical schemes and put hence some more rigor in the construction of such AP-methods.