Stabilized mixed approximation of axisymmetric Brinkman flows

Anaya, Verónica; Mora, David; Reales, Carlos; Ruiz-Baier, Ricardo

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 855-874.

This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order $k\ge 0$, piecewise continuous polynomials of degree $k+1$, and piecewise polynomials of degree $k$, respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions.

Classification : 65N30 65N12 76D07 65N15 65J20
Mots clés : Brinkman equations, axisymmetric domains, augmented mixed finite elements, well-posedness analysis, error estimates

N. Abdellatif and C. Bernardi, A new formulation of the Stokes problem in a cylinder, and its spectral discretization. ESAIM: M2AN 38 (2004) 781–810. | DOI | Numdam | MR | Zbl

N. Abdellatif, N. Chorfi and S. Trabelsi, Spectral discretization of the axisymmetric vorticity, velocity and pressure formulation of the Stokes problem. J. Sci. Comput. 47 (2011) 419–440. | DOI | MR | Zbl

M. Amara, D. Capatina-Papaghiuc, B. Denel and P. Terpolilli, Mixed finite element approximation for a coupled petroleum reservoir model. ESAIM: M2AN 39 (2005) 349–376. | DOI | Numdam | MR | Zbl

V. Anaya, D. Mora and R. Ruiz-Baier, An augmented mixed finite element method for the vorticity-velocity-pressure formulation of the Stokes equations. Comput. Methods Appl. Mech. Engrg. 267 (2013) 261–274. | DOI | MR | Zbl

V. Anaya, D. Mora, G.N. Gatica and R. Ruiz-Baier, An augmented velocity-vorticity-pressure formulation for the Brinkman problem. To appear in Int. J. Numer. Methods Fluids (2015). | MR

V. Anaya, D. Mora, R. Oyarzúa and R. Ruiz-Baier, A priori and a posteriori error analysis for a vorticity-based mixed formulation of the generalized Stokes equations. CI${}^2$MA preprint 2014-20. Available from http://www.ci2ma.udec.cl/publicaciones/prepublicaciones.

S.M. Aouadi, C. Bernardi and J. Satouri, Mortar spectral element discretization of the Stokes problem in axisymmetric domains. Numer. Methods Partial Differ. Eq. 30 (2014) 44–73. | DOI | MR | Zbl

F. Assous, P. Ciarlet and S. Labrunie,Theoretical tools to solve the axisymmetric Maxwell equations. Math. Methods Appl. Sci. 25 (2002) 49–78. | DOI | MR | Zbl

F. Assous, P. Ciarlet, S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method. J. Comput. Phys. 191 (2003) 147–176. | DOI | MR | Zbl

M. Azaiez, A spectral element projection scheme for incompressible flow with application to the unsteady axisymmetric Stokes problem. J. Sci. Comput. 17 (2002) 573–584. | DOI | MR | Zbl

G.R. Barrenechea and F. Valentin, An unusual stabilized finite element method for a generalized Stokes problem. Numer. Math. 92 (2002) 653–677. | DOI | MR | Zbl

T. Barrios, R. Bustinza, G. García and E. Hernández, On stabilized mixed methods for generalized Stokes problem based on the velocity-pseudostress formulation: a priori error estimates. Comput. Methods Appl. Mech. Engrg. 237/240 (2012) 78–87. | DOI | MR | Zbl

Z. Belhachmi, C. Bernardi and S. Deparis, Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math. 105 (2002) 217–247. | DOI | MR | Zbl

Z. Belhachmi, C. Bernardi, S. Deparis and F. Hecht, An efficient discretization of the Navier-Stokes equations in an axisymmetric domain. Part 1: The discrete problem and its numerical analysis. J. Sci. Comput. 27 (2006) 97–110. | DOI | MR | Zbl

A. Bermúdez, C. Reales, R. Rodríguez and P. Salgado, Numerical analysis of a finite element method for the axisymmetric eddy current model of an induction furnace. IMA J. Numer. Anal. 30 (2010) 654–676. | DOI | MR | Zbl

A. Bermúdez, C. Reales, R. Rodríguez and P. Salgado, Mathematical and numerical analysis of a transient eddy current axisymmetric problem involving velocity terms. Numer. Methods Partial Differ. Eq. 28 (2012) 984–1012. | DOI | MR

C. Bernardi and N. Chorfi, Spectral discretization of the vorticity, velocity, and pressure formulation of the Stokes problem. SIAM J. Numer. Anal. 44 (2007) 826–850. | DOI | MR | Zbl

C. Bernardi, M. Dauge and Y. Maday,Spectral Methods for Axisymmetric Domains. Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris (1999). | MR | Zbl

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag, New York (1991). | MR | Zbl

R. Bürger, S. Kumar and R. Ruiz-Baier, Discontinuous finite volume element approximation for a fully coupled Stokes flow-transport problem. CI${}^2$MA preprint 2014-25. Available from http://www.ci2ma.udec.cl/publicaciones/prepublicaciones. | MR

R. Bürger, R. Ruiz-Baier and H. Torres, A stabilized finite volume element formulation for sedimentation–consolidation processes. SIAM J. Sci. Comput. 34 (2012) B265–B289. | DOI | MR | Zbl

J.H. Carneiro De Araujo and V. Ruas, A stable finite element method for the axisymmetric three-field Stokes system. Comput. Methods Appl. Mech. Engrg. 164 (1998) 267–286. | DOI | MR | Zbl

P. Clément, Approximation by finite element functions using local regularisation. RAIRO Modél. Math. Anal. Numér. 9 (1975) 77–84. | Numdam | MR | Zbl

D.M. Copeland, J. Gopalakrishnan and M. Oh, Multigrid in a weighted space arising from axisymmetric electromagnetics. Math. Comput. 79 (2010) 2033–2058. | DOI | MR | Zbl

V.J. Ervin, Approximation of axisymmetric Darcy flow using mixed finite element methods. SIAM J. Numer. Anal. 51 (2013) 1421–1442. | DOI | MR | Zbl

V.J. Ervin, Approximation of coupled Stokes-Darcy flow in an axisymmetric domain. Comput. Methods Appl. Mech. Engrg. 258 (2013) 96–108. | DOI | MR | Zbl

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). | MR | Zbl

G.N. Gatica, A. Márquez, R. Oyarzúa and R. Rebolledo, Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media. Comput. Methods Appl. Mech. Engrg. 270 (2014) 76–112. | DOI | MR | Zbl

G.N. Gatica, L.F. Gatica and A. Márquez, Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126 (2014) 635–677. | DOI | MR | Zbl

J. Gopalakrishnan and M. Oh, Commuting smoothed projectors in weighted norms with an application to axisymmetric Maxwell equations. J. Sci. Comput. 51 (2012) 394–420. | DOI | MR | Zbl

J. Gopalakrishnan and J. Pasciak, The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations. Math. Comput. 75 (2006) 1697–1719. | DOI | MR | Zbl

R. Grauer and T.C. Sideris, Numerical computation of 3D incompressible ideal fluids with swirl. Phys. Rev. Lett. 67 (1991) 3511–3514. | DOI

A. Guardone and L. Vigevano, Finite element/volume solution to axisymmetric conservation laws. J. Comput. Phys. 224 (2007) 489–518. | DOI | MR | Zbl

J. Könnö and R. Stenberg, Numerical computations with $H\left(div\right)-$finite elements for the Brinkman problem. Comput. Geosci. 16 (2012) 139–158. | DOI | MR | Zbl

A. Kufner, Weighted Sobolev Spaces. Wiley, New York (1983). | MR | Zbl

P. Lacoste, Solution of Maxwell equation in axisymmetric geometry by Fourier series decomposition and by use of $H\left(rot\right)$ conforming finite element. Numer. Math. 84 (2000) 577–609. | DOI | MR | Zbl

Y.-J. Lee and H. Li, Axisymmetric Stokes equations in polygonal domains: Regularity and finite element approximations. Comput. Math. Appl. 64 (2012) 3500–3521. | DOI | MR | Zbl

Y.-J. Lee and H. Li, On stability, accuracy, and fast solvers for finite element approximations of the axisymmetric Stokes problem by Hood-Taylor elements. SIAM J. Numer. Anal. 49 (2011) 668–691. | DOI | MR | Zbl

J.-G. Liu and W.-C. Wang, Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows. SIAM J. Numer. Anal. 44 (2006) 2456–2480. | DOI | MR | Zbl

K.A. Mardal, X.-C. Tai and R. Winther, A robust finite element method for Darcy–Stokes flow. SIAM J. Numer. Anal. 40 (2002) 1605–1631. | DOI | MR | Zbl

B. Mercier and G. Raugel, Resolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en $r$, $z$ et séries de Fourier en $\theta$. RAIRO Anal. Numér. 16 (1982) 405–461. | DOI | Numdam | MR | Zbl

G. Pontrelli, Blood flow through an axisymmetric stenosis. Proc. Int. Mech. Engrs. 215 (2001) H01–H10.

K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17 (2007) 215–252. | DOI | MR | Zbl

A. Riaz, M. Hesse and H.A. Tchelepi, Onset of convection in gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548 (2006) 87–111. | DOI | MR

V. Ruas, Mixed finite element methods with discontinuous pressures for the axisymmetric Stokes problem. ZAMM Z. Angew. Math. Mech. 83 (2003) 249–264. | DOI | MR | Zbl

C.G. Speziale, On the advantages of the vorticity-velocity formulations of the equations of fluid dynamics. J. Comput. Phys. 73 (1987) 476–480. | DOI | Zbl

P.N. Tandon and U.V.S. Rana, A new model for blood flow through an artery with axisymmetric stenosis. Int. J. Biomed. Comput. 38 (1995) 257–267. | DOI

P. Vassilevski and U. Villa, A mixed formulation for the Brinkman problem. SIAM J. Numer. Anal. 52 (2014) 258–281. | DOI | MR | Zbl