arXiv subjects

Joel Chaskalovic

Publications and source records attributed to Joel Chaskalovic.

9 recordsLinked to original sources

arXiv2023

Enhancing Interpolation and Approximation Error Estimates Using a Novel Taylor-like Formula

In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the $P_1$-interpolation error estimate. Following the same principles, we also develop a novel numerical scheme for the heat equation that yields a better error estimate compared to the classical implicit finite differences scheme.

math.NA↗

arXiv2023

Improved $P_1$-interpolation error estimates in $W^{1,p}(]0,1[)$: Application to finite element method

Based on a new Taylor-like formula, we derived an improved interpolation error estimate in $W^{1,p}$. We compare it with the classical error estimates based on the standard Taylor formula, and also with the corresponding interpolation error estimate, derived from the mean value theorem. We then assess the improvement in accuracy we can get from this formula, leading to a significant reduction in finite element computation costs.

math.NA↗

arXiv2022

A refined first-order expansion formula in Rn: Application to interpolation and finite element error estimates

The aim of this paper is to derive a refined first-order expansion formula in Rn, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor's formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at $n+1$ equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.

math.NA↗

arXiv2022

A New First Order Taylor-like Theorem With An Optimized Reduced Remainder

This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error estimate and also the trapezoidal quadrature error, in order to assess the gain of accuracy we obtain using this new Taylor-like formula.

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arXiv2020

Numerical validation of probabilistic laws to evaluate finite element error estimates

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.

math.NA↗

arXiv2020

Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

In this paper we propose a new generation of probability laws based on the generalized Beta prime distribution to estimate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1<k_2)$. Since the relative finite element accuracy is usually based on the comparison of the asymptotic speed of convergence when the mesh size $h$ goes to zero, this probability laws highlight that there exists, depending on $h$, cases such that $P_{k_1}$ finite element is more likely accurate than the $P_{k_2}$ one. To confirm this feature, we show and examine on practical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities determined by the probability law. Among others, it validates, when $h$ moves away from zero, that finite element $P_{k_1}$ may produces more precise results than a finite element $P_{k_2}$ since the probability of the event "$P_{k_1}$ is more accurate than $P_{k_2}$" consequently increases to become greater than 0.5. In these cases, $P_{k_2}$ finite elements are more likely overqualified.

math.NA↗

arXiv2020

A probabilistic approach for exact solutions of determinist PDE's as well as their finite element approximations

A probabilistic approach is developed for the exact solution $u$ to a determinist partial differential equation as well as for its associated approximation $u^{(k)}_{h}$ performed by $P_k$ Lagrange finite element. Two limitations motivated our approach: on the one hand, the inability to determine the exact solution $u$ to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation $u^{(k)}_{h}$. We thus fill this knowledge gap by considering the exact solution $u$ together with its corresponding approximation $u^{(k)}_{h}$ as random variables. By way of consequence, any function where $u$ and $u_{h}^{(k)}$ are involved as well. In this paper, we focus our analysis to a variational formulation defined on $W^{m,p}$ Sobolev spaces and the corresponding a priori estimates of the exact solution $u$ and its approximation $u^{(k)}_{h}$ to consider their respective $W^{m,p}-$norm as a random variable, as well as the $W^{m,p}$ approximation error with regards to $P_k$ finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1 < k_2)$.

math.NA↗

arXiv2019

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

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.

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arXiv2018

On generalized binomial laws to evaluate finite element accuracy: toward applications for adaptive mesh refinement

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 $P_k$ and $P_m$, ($k < m$). We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes and the second one is dedicated to a sequence of simplexes which constitute a given mesh. Both of this applications might be relevant for adaptive mesh refinement.

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