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

We derive an explicit k-dependence in (Formula presented.) error estimates for (Formula presented.) Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between (Formula presented.) and (Formula presented.) finite elements, ((Formula presented.)), in terms of (Formula presented.) -norms. We further prove a weak asymptotic relation in (Formula presented.) between these probabilistic laws when difference (Formula presented.) goes to infinity. Moreover, as expected, one finds that (Formula presented.) finite element is surely more accurate than (Formula presented.), for sufficiently small values of the mesh size h. Nevertheless, our results also highlight cases where (Formula presented.) is more likely accurate than (Formula presented.), 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.