Explicit k-dependence for P k finite elements in W m,p error estimates: application to probabilistic laws for accuracy analysis

Abstract

We derive an explicit k-dependence in error estimates for Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between and finite elements, (), in terms of -norms. We further prove a weak asymptotic relation in between these probabilistic laws when difference goes to infinity. Moreover, as expected, one finds that finite element is surely more accurate than , for sufficiently small values of the mesh size h. Nevertheless, our results also highlight cases where is more likely accurate than , for a range of values of h. Hence, this approach brings a new perspective on how to compare two finite elements, which is not limited to the rate of convergence.