From a geometrical interpretation of Bramble Hilbert lemma to a probabilistic distribution for relative finite elements accuracy

Tuesday 22 Jan 2019, 11:00 → 12:00 Europe/Zurich

17/1-007 (CERN)

Speakers:
Prof. Joel Chaskalovic, Sorbonne University. Paris, France
Prof. Franck Assous, Ariel University. Ariel, Israel

Abstract:
The aim of this talk is to provide new perspectives on relative finite elements accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size h goes to zero.

Indeed, for concrete applications, the mesh size h is fixed and one has to decide how to choose a finite element, among other things, regarding its accuracy.

Starting from a geometrical interpretation of the error estimate which can be derived from Bramble-Hilbert lemma, we got two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements Pk and Pm, (k < m).

We established mathematical properties of these probabilistic distributions and we got new insights which, amongst others, show that, despite the usual point of view which claims that Pm finite element are more accurate than Pk ones when k < m, Pk finite element is more likely accurate than Pm when the mesh size h is greater than a critical value h*. 

This new point of view allows us to recommend that for specific situations, like for adaptive refinement meshes for example, Pk finite element would be locally more appropriated than Pm ones, (k < m), as long as one will be able to detect the case h > h*.

More generally, this novel way to evaluate the relative accuracy is not restricted to finite element methods but can be extended to other types of approximations:
Given a class of numerical schemes to approximate solution to ordinary differential equations and their corresponding error estimates, one will be able to order them, not only in terms of asymptotic rate of convergence, but also, for a given mesh size, by evaluating the most probably accurate.

Then, for example, it will be possible to argue why (or why not!) the Runge−Kutta fourth-order method (RK4) would be implemented rather than another simplest one.