On the accuracy of numerical methods for the discretization of anisotropic elliptic problems

In this paper the loss of precision of numerical methods discretizing anisotropic elliptic problems is analyzed. This feature is prominently observed when the coordinates and the mesh are unrelated to the anisotropy direction. This issue is carefully analyzed and related to the asymptotic instability of the discretizations. The investigations carried out within this paper demonstrate that, high order methods commonly implemented to cope with this difficulty, though bringing evident gains, remain for far from optimal and limited to moderate anisotropy strengths. A second issue, related to the reconstruction of the solution discrete parallel gradients, is also addressed. In particular, it is demonstrated that an accurate approximation can hardly be computed from a precise numerical approximation of the solution. A new method is proposed, consisting in introducing an auxiliary variable providing discrete approximations of the parallel gradient with a precision unrelated to the anisotropy strength.