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.
Accepted:
Published online:
Verónica Anaya 1; Bryan Gómez-Vargas 2, 3, 4; David Mora 1, 2; Ricardo Ruiz-Baier 5
@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] Stabilized finite element method for Navier–Stokes equations with physical boundary conditions, Math. Comput., Volume 76 (2007) no. 259, pp. 1195-1217
[2] Vorticity–velocity–pressure formulation for Stokes problem, Math. Comput., Volume 73 (2004) no. 248, pp. 1673-1697
[3] 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] An augmented velocity-vorticity-pressure formulation for the Brinkman equations, Int. J. Numer. Methods Fluids, Volume 79 (2015) no. 3, pp. 109-137
[5] 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] 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] 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] Three–dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal., Volume 34 (1997) no. 2, pp. 664-670
[9] Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, Germany, 2013
[10] 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] 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] 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] 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] Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer-Verlag, Berlin, 1986
[15] Numerical solution of the Navier–Stokes equations using the finite element technique, Comput. Fluids, Volume 1 (1973), pp. 1-28
[16] 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] Effect of variable viscosity on thermohaline convection in a porous medium, J. Hydrol., Volume 57 (1982) no. 1–2, pp. 147-161
[18] 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] Weighted BFBT preconditioner for Stokes flow problems with highly heterogeneous viscosity, SIAM J. Sci. Comput., Volume 39 (2017) no. 5, p. S272-S297
[20] Low-order finite element method for the well-posed bidimensional Stokes problem, IMA J. Numer. Anal., Volume 35 (2015), pp. 427-453
[21] 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] A simple approach to discretize the viscous term with spatially varying (eddy-)viscosity, J. Comput. Phys., Volume 253 (2013), pp. 405-417
[23] 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] A mixed formulation for the Brinkman problem, SIAM J. Numer. Anal., Volume 52 (2014) no. 1, pp. 258-281
[25] 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
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☆ 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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