Minimization of gradient errors of piecewise linear interpolation on simplicial meshes

The paper is devoted to the analysis of optimal simplicial meshes which minimize the gradient error of the piecewise linear interpolation over all conformal simplicial meshes with a fixed number of cells NT. We present theoretical results on asymptotic dependencies of Lp-norms of the gradient error on NT for spaces of arbitrary dimension d. Our analysis is based on a geometric representation of the gradient error of linear interpolation on a simplex and a relaxed saturation assumption. We derive a metric field M p such that a M p-quasi-uniform mesh is quasi-optimal, for arbitrary d and p ∊ ]0, +∞]. Quasi-optimal meshes provide the same asymptotics of the Lp-norm of the gradient error as the optimal meshes.