  This paper deals with the numerical study of a strongly anisotropic heat
equation. The use of standard schemes in this situation leads to poor results,
due to the high anisotropy. Furthermore, the recently proposed
Asymptotic-Preserving method [arXiv:1203.6739] allows to perform simulations
regardless of the anisotropy strength but its application is limited to the
case, where the anisotropy direction is given by a field with all field lines
open. In this paper we introduce a new Asymptotic-Preserving method, which
overcomes those limitations without any loss of precision or increase in the
computational costs. The convergence of the method is shown to be independent
of the anisotropy parameter $0 < \eps <1$, and this for fixed coarse Cartesian
grids and for variable anisotropy directions. The context of this work are
magnetically confined fusion plasmas.
