[Paper Review] Revisiting finite element accuracy by a mixed functional-probabilistic approach.

Joël Chaskalovic, Franck Assous|arXiv (Cornell University)|Mar 26, 2018

Probabilistic and Robust Engineering DesignDecision Sciences12 references3 citations

TL;DR

This paper introduces a mixed functional-probabilistic approach to assess finite element accuracy by modeling the relative accuracy between $P_k$ and $P_m$ elements ($k < m$) as a random variable. Using a geometrical interpretation of the Bramble-Hilbert lemma, it derives two probability distributions that reveal under which mesh size $h$ one element is more likely accurate than the other, with asymptotic behavior analyzed as $m-k \to \infty$. The key contribution is a probabilistic framework that predicts element superiority based on $h$, offering new insights into finite element selection.

ABSTRACT

The aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble-Hilbert lemma, we derive two probability distributions that estimates the relative accuracy, considered as a random variable, between two finite elements $P_k$ and $P_m$, ($k < m$). We establish mathematical properties of these laws, particularly studying their asymptotic relation when the difference $m-k$ goes to infinity. Then, we get new insights which, among others, show that $P_k$ or $P_m$ is more likely accurate than the other, depending on a the value of the mesh size $h$.

Motivation & Objective

• To re-evaluate finite element accuracy through a probabilistic lens, moving beyond deterministic error bounds.
• To model the relative accuracy between two finite elements $P_k$ and $P_m$ ($k < m$) as a random variable.
• To derive probability distributions that estimate the likelihood of one element being more accurate than the other.
• To analyze the asymptotic behavior of these distributions as the polynomial degree difference $m-k$ increases.
• To identify conditions on the mesh size $h$ under which $P_k$ or $P_m$ is more likely accurate.

Proposed method

• Derive a geometrical interpretation of the Bramble-Hilbert lemma to model finite element approximation error in a functional space.
• Construct two probability distributions that estimate the relative accuracy between $P_k$ and $P_m$ elements.
• Use functional analysis to define the random variable representing relative accuracy in terms of approximation norms.
• Analyze the asymptotic properties of these distributions as $m-k \to \infty$, revealing convergence trends.
• Establish mathematical properties of the derived laws, including moments and tail behaviors, to support probabilistic inference.
• Relate the resulting probabilistic model to the mesh size $h$ to determine dominance of one element over the other.

Experimental results

Research questions

• RQ1 How can the relative accuracy between $P_k$ and $P_m$ finite elements be modeled as a random variable?
• RQ2 What probability distributions describe the likelihood of $P_k$ being more accurate than $P_m$?
• RQ3 How does the mesh size $h$ influence the dominance of one element over the other in terms of accuracy?
• RQ4 What is the asymptotic behavior of the accuracy probability distributions as $m-k \to \infty$?
• RQ5 Under what conditions on $h$ is $P_k$ more likely accurate than $P_m$, and vice versa?

Key findings

• The paper derives two probability distributions that model the relative accuracy between $P_k$ and $P_m$ finite elements as a random variable.
• The distributions reveal that the likelihood of $P_k$ or $P_m$ being more accurate depends critically on the mesh size $h$.
• As the difference in polynomial degrees $m-k$ increases, the derived probability laws exhibit specific asymptotic convergence patterns.
• For small $h$, $P_m$ is more likely accurate, while for large $h$, $P_k$ may dominate, depending on the problem's regularity.
• The probabilistic framework provides a new criterion for finite element selection based on $h$, beyond traditional error estimates.