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

## TL;DR

This paper develops explicit $k$-dependence in $W^{m,p}$ error estimates for $P_k$ finite elements and introduces probabilistic laws to compare their accuracy, revealing nuanced insights beyond traditional convergence rates.

## Contribution

It introduces probabilistic laws to compare finite element accuracy and establishes explicit $k$-dependence in $W^{m,p}$ error estimates, offering a new perspective on finite element comparison.

## Key findings

- $P_{k_2}$ is more accurate than $P_{k_1}$ for small mesh sizes.
- Probabilistic laws can show cases where lower-order elements are more likely accurate.
- As $k_2 - k_1$ increases, the probabilistic relation approaches a weak asymptotic limit.

## 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 $D'(R)$ 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.