Comptes Rendus
Numerical analysis
Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 552-560.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.06.006

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
@article{CRMATH_2019__357_6_552_0,
     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},
     publisher = {Elsevier},
     volume = {357},
     number = {6},
     year = {2019},
     doi = {10.1016/j.crma.2019.06.006},
     language = {en},
}
TY  - JOUR
AU  - Verónica Anaya
AU  - Bryan Gómez-Vargas
AU  - David Mora
AU  - Ricardo Ruiz-Baier
TI  - Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 552
EP  - 560
VL  - 357
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2019.06.006
LA  - en
ID  - CRMATH_2019__357_6_552_0
ER  - 
%0 Journal Article
%A Verónica Anaya
%A Bryan Gómez-Vargas
%A David Mora
%A Ricardo Ruiz-Baier
%T Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
%J Comptes Rendus. Mathématique
%D 2019
%P 552-560
%V 357
%N 6
%I Elsevier
%R 10.1016/j.crma.2019.06.006
%G en
%F CRMATH_2019__357_6_552_0
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/

[1] M. Amara; D. Capatina-Papaghiuc; D. Trujillo Stabilized finite element method for Navier–Stokes equations with physical boundary conditions, Math. Comput., Volume 76 (2007) no. 259, pp. 1195-1217

[2] M. Amara; E. Chacón Vera; D. Trujillo Vorticity–velocity–pressure formulation for Stokes problem, Math. Comput., Volume 73 (2004) no. 248, pp. 1673-1697

[3] K. Amoura; M. Azaïez; C. Bernardi; N. Chorfi; S. Saadi Spectral element discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes problem, Calcolo, Volume 44 (2007) no. 3, pp. 165-188

[4] V. Anaya; G.N. Gatica; D. Mora; R. Ruiz-Baier An augmented velocity-vorticity-pressure formulation for the Brinkman equations, Int. J. Numer. Methods Fluids, Volume 79 (2015) no. 3, pp. 109-137

[5] V. Anaya; D. Mora; R. Oyarzúa; R. Ruiz-Baier A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem, Numer. Math., Volume 133 (2016) no. 4, pp. 781-817

[6] C. Bernardi; N. Chorfi Spectral discretization of the vorticity, velocity, pressure formulation of the Stokes problem, SIAM J. Numer. Anal., Volume 44 (2007) no. 2, pp. 826-850

[7] D. Boffi Stability of higher order triangular Hood–Taylor methods for stationary Stokes equations, Math. Models Methods Appl. Sci., Volume 2 (1994) no. 4, pp. 223-235

[8] D. Boffi Three–dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal., Volume 34 (1997) no. 2, pp. 664-670

[9] D. Boffi; F. Brezzi; M. Fortin Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, Germany, 2013

[10] C.L. Chang; B.-N. Jiang An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for the Stokes problem, Comput. Methods Appl. Mech. Eng., Volume 84 (1990) no. 3, pp. 247-255

[11] B. Cockburn; J. Cui An analysis of HDG methods for the vorticity-velocity-pressure formulation of the Stokes problem in three dimensions, Math. Comput., Volume 81 (2012) no. 279, pp. 1355-1368

[12] F. Dubois; M. Salaün; S. Salmon First vorticity-velocity-pressure numerical scheme for the Stokes problem, Comput. Methods Appl. Mech. Eng., Volume 192 (2003) no. 44–46, pp. 4877-4907

[13] G.N. Gatica A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, Heidelberg, New York, Dordrecht, London, 2014

[14] V. Girault; P.A. Raviart Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer-Verlag, Berlin, 1986

[15] P. Hood; C. Taylor Numerical solution of the Navier–Stokes equations using the finite element technique, Comput. Fluids, Volume 1 (1973), pp. 1-28

[16] V. John; K. Kaiser; J. Novo Finite element methods for the incompressible Stokes equations with variable viscosity, ZAMM: J. Appl. Math. Mech., Volume 96 (2015) no. 2, pp. 205-216

[17] P.R. Patil; G. Vaidyanathan Effect of variable viscosity on thermohaline convection in a porous medium, J. Hydrol., Volume 57 (1982) no. 1–2, pp. 147-161

[18] L.E. Payne; J.C. Song; B. Straughan Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, Proc. R. Soc. Lond., Ser. A, Volume 455 (1986), pp. 1-20

[19] J. Rudi; G. Stadler; O. Ghattas Weighted BFBT preconditioner for Stokes flow problems with highly heterogeneous viscosity, SIAM J. Sci. Comput., Volume 39 (2017) no. 5, p. S272-S297

[20] M. Salaün; S. Salmon Low-order finite element method for the well-posed bidimensional Stokes problem, IMA J. Numer. Anal., Volume 35 (2015), pp. 427-453

[21] C.G. Speziale On the advantages of the vorticity-velocity formulations of the equations of fluid dynamics, J. Comput. Phys., Volume 73 (1987) no. 2, pp. 476-480

[22] F.X. Trias; A. Gorobets; A. Oliva A simple approach to discretize the viscous term with spatially varying (eddy-)viscosity, J. Comput. Phys., Volume 253 (2013), pp. 405-417

[23] C.-C. Tsai; S.-Y. Yang On the velocity-vorticity-pressure least-squares finite element method for the stationary incompressible Oseen problem, J. Comput. Appl. Math., Volume 182 (2005) no. 1, pp. 211-232

[24] P.S. Vassilevski; U. Villa A mixed formulation for the Brinkman problem, SIAM J. Numer. Anal., Volume 52 (2014) no. 1, pp. 258-281

[25] J. Woodfield; M. Alvarez; B. Gómez-Vargas; R. Ruiz-Baier Stability and finite element approximation of phase change models for natural convection in porous media, J. Comput. Appl. Math., Volume 360 (2019), pp. 117-137

Cited by 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.

Comments - Policy