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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, Comment: 15 pages, 11 figures

The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements Pk and Pm, k < m. We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods.

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 recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements $P_k$ and $P_m$, ($k < m$). Then, we analyze the asymptotic relation between these two probabilistic laws when the difference $m-k$ goes to infinity. New insights which qualified the relative accuracy in the case of high order finite elements are correspondingly obtained., Comment: 20 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1803.09547

We derive an explicit k-dependence in Wm,p error estimates for Pk Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between Pk1 and Pk2 finite eleme...

In this paper, we propose a new approach based on data mining techniques and probabilistic models to compare and analyze finite element results of partial differential equations. We focus on the numerical errors produced by linear and quadratic finite element approximations. We first show how error estimates contain a kind of numerical uncertainty in their evaluation, which may influence and even damage the precision of finite element numerical results. A model problem, derived from an elliptic approximate Vlasov-Maxwell system, is then introduced. We define some variables as physical predictors, and we characterize how they influence the odds of the linear and quadratic finite elements to be locally "same order" accurate. Beyond this example, this approach proposes a method to compare, between several approximation methods, the accuracy of numerical results.

The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size h goes to zero. Starting from a geometrical reading of the error estimate due to Bramble-Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements \(P_k\) and \(P_m\), (\(k < m\)). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that \(P_k\) or \(P_m\) is more likely accurate than the other, depending on the value of the mesh size h.

Rolle’s theorem, and therefore, Lagrange and Taylor’s theorems are responsible for the inability to determine precisely the error estimate of numerical methods applied to partial differential equations. Basically, this comes from the existence of a non unique unknown point which appears in the remainder of Taylor’s expansion. In this paper we consider the case of finite elements method. We show in detail how Taylor’s theorem gives rise to indeterminate constants in the a priori error estimates. As a consequence, we highlight that classical conclusions have to be reformulated if one considers local error estimate. To illustrate our purpose, we consider the implementation of P 1 and P 2 finite elements method to solve Vlasov–Maxwell equations in a paraxial configuration. If the Bramble–Hilbert theorem claims that global error estimates for finite elements P 2 are “better” than the P 1 ones, we show how data mining techniques are powerful to identify and to qualify when and where local numerical results of P 1 and P 2 are equivalent.

This paper deals with the applications of data mining techniques in the evaluation of numerical solutions of Vlasov–Maxwell models. This is part of the topic of characterizing the model and approximation errors via learning techniques. We give two examples of application. The first one aims at comparing two Vlasov–Maxwell approximate models. In the second one, a scheme based on data mining techniques is proposed to characterize the errors between a P 1 and a P 2 finite element Particle-In-Cell approach. Beyond these examples, this original approach should operate in all cases where intricate numerical simulations like for the Vlasov–Maxwell equations take a central part.

In this Note, we propose a new methodology based on exploratory data mining techniques to evaluate the errors due to the description of a given real system. First, we decompose this description error into four types of sources. Then, we construct databases of the entire information produced by different numerical approximation methods, to assess and compare the significant differences between these methods, using techniques like decision trees, Kohonenʼs cards, or neural networks. As an example, we characterize specific states of the real system for which we can locally appreciate the accuracy between two kinds of finite elements methods. In this case, this allowed us to precise the classical Bramble–Hilbert theorem that gives a global error estimate, whereas our approach gives a local error estimate.

This paper proposes a statistical and probabilistic approach to compare and analyze the errors of two different approximation methods. We introduce the principle of numerical uncertainty in such a process, and we illustrate it by considering the discretization difference between two different approximation orders, e.g., first and second order Lagrangian finite element. Then, we derive a probabilistic approach to define and to qualify equivalent results. We illustrate our approach on a model problem on which we built the two above mentioned finite element approximations. We consider some variables as physical “predictors”, and we characterize how they influence the odds of the approximation methods to be locally “same order accurate”.

Developments in numerical analysis in the twentieth century generated many methods for obtaining approximate solutions to partial differential equations. Compared with the finite-difference method, spectral methods, the finite-volume method, or the singularity method, the successes of the finite-element method are unquestionable, and its supremacy is quite justifiably acclaimed.

The aim in this chapter will be to set up and discuss, in a one-dimensional context, variational formulations and the implementation of a Lagrange finite-element analysis.