Optimal Meshes for Finite Elements of Arbitrary Order

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.