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The concern of the present work is the introduction of a very efficientAsymptotic Preserving scheme for the resolution of highly anisotropic diffusionequations. The characteristic features of this scheme are the uniformconvergence with respect to the anisotropy parameter $0<\eps <<1$, theapplicability (on cartesian grids) to cases of non-uniform and non-alignedanisotropy fields $b$ and the simple extension to the case of a non-constantanisotropy intensity $1/\eps$. The mathematical approach and the numericalscheme are different from those presented in the previous work [Degond et al.(2010), arXiv:1008.3405v1] and its considerable advantages are pointed out.

An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition