Anisotropic meshes are known to be well suited for problems which exhibit anisotropic
solution features. Define an appropriate metric tensor and design an ecient algorithm
for anisotropic mesh generation are two important aspects of the anisotropic mesh
methodology. In this talk, we are concerned with the perfect metric tensor for use in
anisotropic mesh generation for anisotropic elliptic problem. We provide an algorithm
to fast generate anisotropic meshes under the given metric tensor. We show that the
inverse of the anisotropic diffusion matrix of the anisotropic elliptic problem is a
pefect matched metric for the anisotropic mesh in three aspects: better discrete
algebraic systems, more accurate finite element solution and superconvergence on the
mesh nodes. Various numerical examples demonstrating the effectiveness are presented.