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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 measurethe 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 relationin $D'(R)$ between these probabilistic laws when difference $k_2-k_1$ goes toinfinity. 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 themesh 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 finiteelements, which is not limited to the rate of convergence.

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