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 $W^{m,p}$ 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 $D^{\prime }\left(\mathbb{R}\right)$ 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 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 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.

Keywords:

• Error estimates
• finite elements
• Céa lemma
• Bramble–Hilbert lemma
• Banach Sobolev spaces
• probabilistic laws

2010 Mathematics Subject Classifications:

• 65N15
• 65N75
• 65N30

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Joël Chaskalovic  http://orcid.org/0000-0003-1263-5313

Notes

1 The space of functions locally integrable for any compact K of $\mathbb{R}$.