Error estimates of the discretization of linear parabolic equations on general nonconforming spatial grids

Abdallah Bradji; Jürgen Fuhrmann

doi:10.1016/j.crma.2010.09.020

Numerical Analysis

Error estimates of the discretization of linear parabolic equations on general nonconforming spatial grids
[Des estimations d'erreurs pour la discrétisation des équations paraboliques sur une classe générale multidimensionnelle de maillages non conformes]

Présenté par : Philippe G. Ciarlet

Abdallah Bradji ¹ ; Jürgen Fuhrmann ²

¹ Department of Mathematics, University of Annaba-Algeria, 23000 Annaba, Algeria
² Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1119-1122.

Résumé
Abstract

Une classe assez générale de maillages non conformes a été récemment utilisée pour approximer les équations stationnaires de diffusion hétérogène anisotrope pour toute dimension d'espace. Le but de ce travail est d'obtenir des estimations d'erreur pour la discrétisation des équations paraboliques sur cette classe générale de maillages. On présente un schéma implicite où la condition initiale a été discrétisée en utilisant une “projection orthogonale” de la condition initiale. Nous fournissons des estimations d'erreur en normes discrètes de $L^{\infty} (0, T; H_{0}^{1} (Ω))$ et $W^{1, \infty} (0, T; L^{2} (Ω))$ .

A general class of nonconforming meshes has been recently used to approximate stationary anisotropic heterogeneous diffusion problems in any space dimensions. The aim of the present work is to deal with some error estimates of the discretization of parabolic equations on this general class of meshes in several space dimensions. We present an implicit scheme based on an orthogonal projection of the exact initial function. We provide error estimates in discrete norms $L^{\infty} (0, T; H_{0}^{1} (Ω))$ and $W^{1, \infty} (0, T; L^{2} (Ω))$ .

Reçu le : 2010-03-05
Accepté le : 2010-09-24
Publié le : 2010-10-01

DOI : 10.1016/j.crma.2010.09.020

Affiliations des auteurs :

Abdallah Bradji ¹ ; Jürgen Fuhrmann ²

¹ Department of Mathematics, University of Annaba-Algeria, 23000 Annaba, Algeria
² Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

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     title = {Error estimates of the discretization of linear parabolic equations on general nonconforming spatial grids},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1119--1122},
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     year = {2010},
     doi = {10.1016/j.crma.2010.09.020},
     language = {en},
}

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Abdallah Bradji; Jürgen Fuhrmann. Error estimates of the discretization of linear parabolic equations on general nonconforming spatial grids. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1119-1122. doi : 10.1016/j.crma.2010.09.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.020/

Bibliographie
Cité par

[1] R. Eymard; T. Gallouët; R. Herbin Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes, IMA Journal of Numerical Analysis (2009) (Advance Access published on June 16) | DOI

[1] A. Bradji Some simples error estimates for finite volume approximation of parabolic equations, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 9–10, pp. 571-574

Fayssal Benkhaldoun; Abdallah Bradji $L^{\infty} (H_{d i v}) \times W^{1, \infty} (L^{2})$ -norm superconvergence of mixed finite element schemes applied to multidimensional parabolic equations, Communications on Analysis and Computation, Volume 3 (2025) no. 0, p. 40 | DOI:10.3934/cac.2025003
Abdallah Bradji; Daniel Lesnic Steady-state inhomogeneous diffusion with generalized oblique boundary conditions, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 57 (2023) no. 5, p. 2701 | DOI:10.1051/m2an/2023063
Abdallah Bradji; Moussa Ziggaf A Convergence Result of a Linear SUSHI Scheme Using Characteristics Method for a Semi-linear Parabolic Equation, Advances in High Performance Computing, Volume 902 (2021), p. 452 | DOI:10.1007/978-3-030-55347-0_38
Fayssal Benkhaldoun; Abdallah Bradji Convergence Analysis of a Finite Volume Gradient Scheme for a Linear Parabolic Equation Using Characteristic Methods, Large-Scale Scientific Computing, Volume 11958 (2020), p. 566 | DOI:10.1007/978-3-030-41032-2_65
Abdallah Bradji; Jürgen Fuhrmann On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems, Computational and Applied Mathematics, Volume 37 (2018) no. 3, p. 2533 | DOI:10.1007/s40314-017-0463-8
Timothy Barth; Raphaèle Herbin; Mario Ohlberger Finite Volume Methods: Foundation and Analysis, Encyclopedia of Computational Mechanics Second Edition (2017), p. 1 | DOI:10.1002/9781119176817.ecm2010
Abdallah Bradji An analysis for the convergence order of gradient schemes for semilinear parabolic equations, Computers Mathematics with Applications, Volume 72 (2016) no. 5, p. 1287 | DOI:10.1016/j.camwa.2016.06.031
Abdallah Bradji A full analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation, Journal of Mathematical Analysis and Applications, Volume 416 (2014) no. 1, p. 258 | DOI:10.1016/j.jmaa.2014.02.043
Abdallah Bradji; Jürgen Fuhrmann Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Applications of Mathematics, Volume 58 (2013) no. 1, p. 1 | DOI:10.1007/s10492-013-0001-y
Abdallah Bradji An analysis of a second-order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes, Applied Mathematics and Computation, Volume 219 (2013) no. 11, p. 6354 | DOI:10.1016/j.amc.2012.12.050
Abdallah Bradji Some Abstract Error Estimates of a Finite Volume Scheme for the Wave Equation on General Nonconforming Multidimensional Spatial Meshes, Finite Volumes for Complex Applications VI Problems Perspectives, Volume 4 (2011), p. 175 | DOI:10.1007/978-3-642-20671-9_19

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