Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

Verónica Anaya; Bryan Gómez-Vargas; David Mora; Ricardo Ruiz-Baier

doi:10.1016/j.crma.2019.06.006

Numerical analysis

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
[Intégration de la viscosité variable dans des formulations en tourbillon pour les équations de Brinkman]

Présenté par : the Editorial Board

Verónica Anaya ¹ ; Bryan Gómez-Vargas ^2,^3,⁴ ; David Mora ^1,² ; Ricardo Ruiz-Baier ⁵

¹ GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
² CI
³ Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
⁴ Sección de Matemática, Sede de Occidente, Universidad de Costa Rica, San Ramón de Alajuela, Costa Rica
⁵ Mathematical Institute, University of Oxford, A. Wiles Building, Woodstock Road, Oxford OX2 6GG, UK

Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 552-560.

Résumé
Abstract

Dans cette note, on introduit une formulation non symétrique des éléments finis mixtes pour les équations de Brinkman écrites en fonction de la vitesse, du tourbillon et de la pression du fluide, avec viscosité variable. L'analyse de la résolubilité est effectuée à l'aide de la théorie classique de Babuška–Brezzi, et on remarque que n'importe quelle paire d'éléments finis stables pour l'approximation de la vitesse et de la pression pour le problème de Stokes peut être couplée à un espace discret d'ordre arbitraire pour l'approximation du tourbillon. On établit ensuite des bornes d'erreur a priori optimales, qui sont ainsi confirmées par des exemples numériques.

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity, and pressure with non-constant viscosity. The analysis is performed by the classical Babuška–Brezzi theory, and we state that any inf–sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates, which are further confirmed through computational examples.

Reçu le : 2019-04-27
Accepté le : 2019-06-10
Publié le : 2019-06-01

DOI : 10.1016/j.crma.2019.06.006

Affiliations des auteurs :

Verónica Anaya ¹ ; Bryan Gómez-Vargas ^{2, 3, 4} ; David Mora ^{1, 2} ; Ricardo Ruiz-Baier ⁵

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     author = {Ver\'onica Anaya and Bryan G\'omez-Vargas and David Mora and Ricardo Ruiz-Baier},
     title = {Incorporating variable viscosity in vorticity-based formulations for {Brinkman} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {552--560},
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     year = {2019},
     doi = {10.1016/j.crma.2019.06.006},
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Verónica Anaya; Bryan Gómez-Vargas; David Mora; Ricardo Ruiz-Baier. Incorporating variable viscosity in vorticity-based formulations for Brinkman equations. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 552-560. doi : 10.1016/j.crma.2019.06.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.006/

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Cité par Sources :

^☆ Funding: CONICYT-Chile through FONDECYT project 11160706, through Becas-Chile Programme for foreign students and through the project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal.

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