Numerical validation of probabilistic laws to evaluate finite element error estimates

TL;DR

This paper numerically validates a probabilistic framework for comparing finite element methods, demonstrating practical cases where lower-order elements outperform higher-order ones in accuracy.

Contribution

It introduces a numerical validation of a recent probabilistic approach to evaluate finite element error estimates, emphasizing practical implications.

Findings

Finite element $P_k$ can be more accurate than $P_m$ in certain cases

Probabilistic error estimates provide insights beyond classical asymptotic results

Highlights caution in comparing numerical methods solely based on traditional error bounds

Abstract

We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements $P_k$ and $P_m, (k<m)$. In particular, we show practical cases where finite element $P_{k}$ gives more accurate results than finite element $P_{m}$. This illustrates the theoretical probabilistic framework we recently derived in order to evaluate the actual accuracy. This also highlights the importance of the extra caution required when comparing two numerical methods, since the classical results of error estimates concerns only the asymptotic convergence rate.