scholarly journals NUMERICAL VALIDATION OF PROBABILISTIC LAWS TO EVALUATE FINITE ELEMENT ERROR ESTIMATES

2021 ◽  
Vol 26 (4) ◽  
pp. 684-695
Author(s):  
Jöel Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Error Estimates ◽  
Probabilistic Approach ◽  
Asymptotic Convergence ◽  
Probabilistic Framework ◽  
Numerical Validation ◽  
Element Error ◽  
Asymptotic Convergence Rate ◽  
Actual Accuracy

We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements Pk and Pm,(k < m). In particular, we show practical cases where finite element Pk gives more accurate results than finite element Pm. 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.

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe 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 qualify the relative accuracy in the case of high order finite elements are also obtained.

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

2020 ◽  
Vol 20 (2) ◽  
pp. 361-378
Author(s):  
Tamal Pramanick ◽  
Rajen Kumar Sinha
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Error Estimates ◽  
Finite Element Approximation ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Dirichlet Problems ◽  
Element Mesh ◽  
Polygonal Domains ◽  
Numer Math

AbstractThe purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the {L^{\infty}(H^{1})}-norm and {L^{\infty}(L^{2})}-norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

scholarly journals Finite element error estimates for subsonic flow

1987 ◽  
Vol 29 (1) ◽  
pp. 88-102 ◽  
Author(s):  
S. S. Chow ◽  
G. F. Carey
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Subsonic Flow ◽  
Full Potential ◽  
Linear Potential ◽  
Element Analysis ◽  
Subsonic Flows ◽  
Potential Equation ◽  
Dual Formulation ◽  
Element Error

AbstractError estimates are derived for a finite element analysis of plane steady subsonic flows described by the full potential equation. The analysis is based on the use of the theory of variational inequalities to accomodate the subsonic flow constraint and leads to a suboptimal estimate relative to that obtained for linear potential flow. We then consider an alternative dual formulation of the problem and obtain an optimal estimate subject to reasonable regularity assumptions.

scholarly journals Finite element error estimates on the boundary with application to optimal control

2014 ◽  
Vol 84 (291) ◽  
pp. 33-70 ◽  
Author(s):  
Thomas Apel ◽  
Johannes Pfefferer ◽  
Arnd Rösch
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
Element Error

scholarly journals Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0 ◽  
Author(s):  
Dominik Hafemeyer ◽  
◽  
Florian Mannel ◽  
Ira Neitzel ◽  
Boris Vexler ◽  
...  
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
One Dimensional ◽  
Bv Functions ◽  
Elliptic Optimal Control ◽  
Element Error

scholarly journals A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 349
Author(s):  
Joël Chaskalovic
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Finite Element ◽  
Finite Elements ◽  
A Priori Estimates ◽  
Probabilistic Approach ◽  
Approximation Error ◽  
Random Variables ◽  
Partial Differential

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation uh(k) performed by Pk Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u relative 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 uh(k). We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation uh(k) as random variables. By a method of consequence, any function where u and uh(k) are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on Wm,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation uh(k) in order to consider their respective Wm,p-norm as a random variable, as well as the Wm,p approximation error with regards to Pk finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements Pk1 and Pk2,(k1<k2).

Sign in / Sign up

Export Citation Format

Share Document