Numerical validation of probabilistic laws to evaluate finite element error estimates
Abstract
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.
Keyword : numerical validation, error estimates, finite elements, Bramble-Hilbert lemma, probability
How to Cite
Chaskalovic, J., & Assous, F. (2021). Numerical validation of probabilistic laws to evaluate finite element error estimates. Mathematical Modelling and Analysis, 26(4), 684-695. https://doi.org/10.3846/mma.2021.14079
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Chaskalovic and F. Assous. A new mixed functional-probabilistic approach for finite element accuracy. Computational Methods in Applied Mathematics, 20(4):799–813, 2020. https://doi.org/10.1515/cmam-2019-0089
J. Chaskalovic and F. Assous. A new probabilistic interpretation of the BrambleHilbert lemma. Computational Methods in Applied Mathematics, 20(1):79–87, 2020. https://doi.org/10.1515/cmam-2018-0270
J. Chaskalovic and F. Assous. Explicit k-dependence for pk finite elements in wm,p error estimates: application to probabilistic laws for accuracy analysis. Applicable Analysis, 100(13):2825–2843, 2021. https://doi.org/10.1080/00036811.2019.1698727
P.G. Ciarlet and P.A. Raviart. General Lagrange and Hermite interpolation in rn with applications to finite element methods. Archive for Rational Mechanics and Analysis, 46(3):177–199, 1972. https://doi.org/10.1007/BF00252458
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F. Hecht. New development in freefem++. Journal of Numerical Mathematics, 20(3–4):251–266, 2012. https://doi.org/10.1515/jnum-2012-0013
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E. Novak. Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/BFb0079792
E. Novak. Stochastic error bounds for some nonlinear problems in numerical analysis, pp. 503–506. Academic Press, 01 1989.
P.A. Raviart and J.M. Thomas. Introduction á l’analyse numérique des équations aux dérivées partielles. Lecture Notes in Mathematics. Masson, 1982. https://doi.org/10.1007/BFb0079792
R.J. Rossi. Mathematical Statistics: An introduction to likelihood based Inference. John Wiley & Sons, New York, 2018. https://doi.org/0.1002/9781118771075
C. Runge. Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Zeitschrift für Mathematik und Physik, 46, 1901.
G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Automatic Computation. Prentice-Hall, 1973.
J.S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. exts in Applied Mathematics. Springer, New York, 2008. https://doi.org/10.1007/978-0-387-72067-8
D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39(5):1749–1779, 2002. https://doi.org/10.1137/S0036142901384162
F. Assous and J. Chaskalovic. Data mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultrarelativistic particles. Journal of Computational Physics, 230(12):4811–4827, 2011. https://doi.org/10.1016/j.jcp.2011.03.005
F. Assous and J. Chaskalovic. Error estimate evaluation in numerical approximations of partial differential equations: A pilot study using data mining methods. Comptes Rendus Mécanique, 341(3):304–313, 2013. https://doi.org/10.1016/j.crme.2013.01.002
I. Babuška. Error-bounds for finite element method. Numerische Mathematik, 16(4):322–333, 1971. https://doi.org/10.1007/BF02165003
J.H. Bramble and S.R. Hilbert. Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM Journal on Numerical Analysis, 7(1):112–124, 1970. https://doi.org/10.1137/0707006
J. Chaskalovic. Mathematical and numerical methods for partial differential equations. Springer Verlag, 2013. https://doi.org/10.1007/978-3-319-03563-5
J. Chaskalovic and F. Assous. Data mining and probabilistic models for error estimate analysis of finite element method. Mathematics and Computers in Simulation, 129:50–68, 2016. https://doi.org/10.1016/j.matcom.2016.03.013
J. Chaskalovic and F. Assous. Probabilistic approach to characterize quantitative uncertainty in numerical approximations. Mathematical Modelling and Analysis, 22(1):106–120, 2017. https://doi.org/10.1080/00036811.2019.1698727
J. Chaskalovic and F. Assous. A new mixed functional-probabilistic approach for finite element accuracy. Computational Methods in Applied Mathematics, 20(4):799–813, 2020. https://doi.org/10.1515/cmam-2019-0089
J. Chaskalovic and F. Assous. A new probabilistic interpretation of the BrambleHilbert lemma. Computational Methods in Applied Mathematics, 20(1):79–87, 2020. https://doi.org/10.1515/cmam-2018-0270
J. Chaskalovic and F. Assous. Explicit k-dependence for pk finite elements in wm,p error estimates: application to probabilistic laws for accuracy analysis. Applicable Analysis, 100(13):2825–2843, 2021. https://doi.org/10.1080/00036811.2019.1698727
P.G. Ciarlet and P.A. Raviart. General Lagrange and Hermite interpolation in rn with applications to finite element methods. Archive for Rational Mechanics and Analysis, 46(3):177–199, 1972. https://doi.org/10.1007/BF00252458
W. Dörfler and R.H. Nochetto. Small data oscillation implies the saturation assumption. Numerische Mathematik, 91(1):1–12, 2002. https://doi.org/10.1007/s002110100321
J.F. Epperson. On the Runge example. The American Mathematical Monthly, 94(4):329–341, 1987. https://doi.org/10.1080/00029890.1987.12000642
F. Hecht. New development in freefem++. Journal of Numerical Mathematics, 20(3–4):251–266, 2012. https://doi.org/10.1515/jnum-2012-0013
M. Le Jeune. Théorie de l’estimation paramétrique ponctuelle. Springer, Paris, 2010. https://doi.org/10.1007/978-2-8178-0157-5_6
P. Lasaint and P.A. Raviart. On a finite element method for solving the neutron transport equation. In Carl de Boor(Ed.), Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–123. Academic Press, 1974. https://doi.org/10.1016/B978-0-12-208350-1.50008-X
W.F. Mitchell. How high a degree is high enough for high order finite elements? Procedia Computer Science, 51:246–255, 2015. https://doi.org/10.1016/j.procs.2015.05.235
E. Novak. Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/BFb0079792
E. Novak. Stochastic error bounds for some nonlinear problems in numerical analysis, pp. 503–506. Academic Press, 01 1989.
P.A. Raviart and J.M. Thomas. Introduction á l’analyse numérique des équations aux dérivées partielles. Lecture Notes in Mathematics. Masson, 1982. https://doi.org/10.1007/BFb0079792
R.J. Rossi. Mathematical Statistics: An introduction to likelihood based Inference. John Wiley & Sons, New York, 2018. https://doi.org/0.1002/9781118771075
C. Runge. Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Zeitschrift für Mathematik und Physik, 46, 1901.
G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Automatic Computation. Prentice-Hall, 1973.