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

Joel Chaskalovic, Franck Assous

Published 2019-01-21Version 1

An explicit $k-$dependency in $W^{m,p}$ error estimates is proved to derive two probabilistic laws which evaluate the relative accuracy between Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1 < k_2)$. Thanks to these estimates we show a weak asymptotic relation in ${\cal D}'(\mathbb{R})$ between these probabilistic laws when $k_2-k_1$ goes to infinity. In this case, we get that $P_{k_2}$ finite element is surely more accurate than $P_{k_1}$, for any value of the mesh size $h$. This brings a complementary perspective regarding the classical way of comparing two finite elements, which is usually limited to the asymptotic rate of convergence, when $h$ goes to zero. Moreover, we also get new insights which highlight cases such that $P_{k_1}$ finite element is more likely accurate than $P_{k_2}$ one, for a range of specific values of $h$.