Explicit k-dependency 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 $W^{m,p}$ error estimates for $P_k$ Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between $P_{k_1}$ and $P_{k_2}$ finite elements ($k_1 < k_2$) in terms of $W^{m,p}$-norms. We further prove a weak asymptotic relation in $D'(R)$ between these probabilistic laws when difference $k_2-k_1$ goes to infinity. Moreover, as expected, one finds that $P_{k_2}$ finite element is {\em surely more accurate} than $P_{k_1}$, for sufficiently small values of the mesh size $h$. Nevertheless, our results also highlight cases where $P_{k_1}$ is {\em more likely accurate} than $P_{k_2}$, 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.