The authors discuss an anisotropic interpolation method and apply it to finite element techniques. To maintain a certain level of accuracy in the solutions of partial differential equations, it is often necessary near anisotropic structures like edges and boundary layers to construct finite element meshes with different mesh sizes in different directions (anisotropic). Given this motivation, the authors present interpolation estimates that may be useful to programmers writing finite element codes.

The paper presents some conditions, based on Bramble-Hilbert theory, for deciding whether an anisotropic estimate holds for a given finite element. The difference between interpolation error and finite element approximation error is discussed, and some results for rectangular elements are presented. The results are illustrated by an application to the finite element approximation of an elliptic equation on domains with edges.