Given a function f defined on a bidimensional bounded domain and a positive
integer N, we study the properties of the triangulation that minimizes the
distance between f and its interpolation on the associated finite element
space, over all triangulations of at most N elements. The error is studied in
the Lp norm and we consider Lagrange finite elements of arbitrary polynomial
degree m-1. We establish sharp asymptotic error estimates as N tends to
infinity when the optimal anisotropic triangulation is used, recovering the
earlier results on piecewise linear interpolation, an improving the results on
higher degree interpolation. These estimates involve invariant polynomials
applied to the m-th order derivatives of f. In addition, our analysis also
provides with practical strategies for designing meshes such that the
interpolation error satisfies the optimal estimate up to a fixed multiplicative
constant. We partially extend our results to higher dimensions for finite
elements on simplicial partitions of a domain of arbitrary dimension.

Key words : anisotropic finite elements, adaptive meshes, interpolation,
nonlinear approximation.