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Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem. (English) Zbl 1381.76158

Summary: In this article, we propose and analyze a new mixed variational formulation for the stationary Boussinesq problem. Our method, which uses a technique previously applied to the Navier-Stokes equations, is based first on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. Next, the pressure is eliminated, and an augmented approach for the fluid flow, which incorporates Galerkin-type terms arising from the constitutive and equilibrium equations, and from the Dirichlet boundary condition, is coupled with a primal-mixed scheme for the main equation modeling the temperature. In this way, the only unknowns of the resulting formulation are given by the aforementioned nonlinear pseudostress, the velocity, the temperature, and the normal derivative of the latter on the boundary. An equivalent fixed-point setting is then introduced and the corresponding classical Banach Theorem, combined with the Lax-Milgram Theorem and the Babuška-Brezzi theory, are applied to prove the unique solvability of the continuous problem. In turn, the Brouwer and the Banach fixed-point theorems are used to establish existence and uniqueness of solution, respectively, of the associated Galerkin scheme. In particular, Raviart-Thomas spaces of order \(k\) for the pseudostress, continuous piecewise polynomials of degree \(\leq k+1\) for the velocity and the temperature, and piecewise polynomials of degree \(\leq k\) for the boundary unknown become feasible choices. Finally, we derive optimal a priori error estimates, and provide several numerical results illustrating the good performance of the augmented mixed-primal finite element method and confirming the theoretical rates of convergence.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Software:

FreeFem++; UMFPACK
Full Text: DOI

References:

[1] K.Allali, A priori and a posteriori error estimates for Boussinesq equations, Int J Numer Anal Model2 (2005), 179-196. · Zbl 1143.76419
[2] C.Bernardi, B.Métivet, B.Pernaud‐Thomas, Couplage des équations de Navier‐Stokes et de la chaleur: le modèle et son approximation par éléments finis, ESAIM: Math Model Numer Anal29 (1995), 871-921. · Zbl 0839.76016
[3] A.Çibik, K.Songül, A projection‐based stabilized finite element method for steady‐state natural convection problem, J Math Anal Appl381 (2011), 469-484. · Zbl 1331.76066
[4] M.Farhloul, S.Nicaise and L.Paquet, A mixed formulation of Boussinesq equations: analysis of nonsingular solutions, Math Comput69 (2000), 965-986. · Zbl 0965.76080
[5] M.Farhloul, S.Nicaise and L.Paquet, A refined mixed finite element method for the Boussinesq equations in polygonal domains, IMA J Numer Anal21 (2001), 525-551. · Zbl 0992.65130
[6] G.Lube, T.Knopp, G.Rapin, R.Gritzki and M.Rösler, Stabilized finite element methods to predict ventilation efficiency and thermal comfort in buildings, Int J Numer Methods Fluids57 (2008), 1269-1290. · Zbl 1338.76058
[7] R.Oyarzúa, T.Qin and D.Schötzau, An exactly divergence‐free finite element method for a generalized Boussinesq problem, IMA J Numer Anal34 (2014), 1104-1135. · Zbl 1301.76052
[8] M, Tabata, D.Tagami, Error estimates of finite element methods for nonstationary thermal convection problems with temperature‐dependent coefficients, Numer Math100 (2005), 351-372. · Zbl 1082.65090
[9] J.Camaño, R.Oyarzua and G.Tierra, Analysis of an augmented mixed‐FEM for the Navier Stokes Problem, Preprint 2014‐33, Centro de Investigación en Ingeniería Matemática (CI^2MA), Universidad de Concepción, Chile, submitted to Math Comp.
[10] Z.Cai and Y.Wang, Pseudostress‐velocity formulation for incompressible Navier‐Stokes equations, Int J Numer Methods Fluids63 (2010), 341-356. · Zbl 1352.76043
[11] Z.Cai, C.Wang and S.Zhang, Mixed finite element methods for incompressible flow: stationary Navier‐Stokes equations, SIAM J Numer Anal48 (2010), 79-94. · Zbl 1410.76160
[12] Z.Cai and S.Zhang, Mixed methods for stationary Navier‐Stokes equations based on pseudostress‐pressure‐velocity formulation, Math Comput81 (2012), 1903-1927. · Zbl 1307.76050
[13] J. S.Howell and N.Walkington, Dual mixed finite element methods for the Navier Stokes equations, ESAIM: Math Model Numer Anal47 (2013), 789-805. · Zbl 1266.76029
[14] Z.Cai, Ch.Tong, P. S.Vassilevski and Ch.Wang, Mixed finite element methods for incompressible flow: stationary Stokes equations, Numer Methods Partial Differential Equations26 (2009), 957-978. · Zbl 1267.76059
[15] L.Figueroa, G. N.Gatica and A.Márquez, Augmented mixed finite element methods for the stationary Stokes equations, SIAM J Sci Comput31 (2008), 1082-1119. · Zbl 1251.74032
[16] G. N.Gatica, L. F.Gatica and A.Márquez, Augmented mixed finite element methods for a vorticity‐based velocity-pressure-stress formulation of the Stokes problem in 2D, Int J Numer Methods Fluids67 (2011), 450-477. · Zbl 1316.76038
[17] G. N.Gatica, A.Márquez and M. A.Sánchez, Analysis of a velocity‐pressure‐pseudostress formulation for incompressible flow, Comput Methods Appl Mech Eng199 (2010), 1064-1079. · Zbl 1227.76030
[18] J. S.Howell, Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace‐free velocity gradients, J Comput Appl Math231 (2009), 780-792. · Zbl 1167.76021
[19] F.Brezzi and M.Fortin, Mixed and hybrid finite element methods, Springer‐Verlag, New York, 1991. · Zbl 0788.73002
[20] G. N.Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowing \(\text{RT}_0\) -P_1-P_0 approximations, ESAIM: Math Model Numer Anal40 (2006), 1-28. · Zbl 1330.74155
[21] G. N.Gatica, An augmented mixed finite element method for linear elasticity with non‐homogeneous Dirichlet conditions, Electron Trans Numer Anal26 (2007), 421-438. · Zbl 1170.74049
[22] M.Alvarez, G. N.Gatica, and R.Ruiz‐Baier, An augmented mixed‐primal finite element method for a coupled flow‐transport problem, ESAIM: Math Model Numer Anal, http://dx.doi.org/10.1051/m2an/2015015. · Zbl 1329.76157
[23] G. N.Gatica, A simple introduction to the mixed finite element method: theory and applications, Springer Briefs in Mathematics, Springer, Cham, 2014. · Zbl 1293.65152
[24] R. A.Adams and J. J. F.Fournier, Sobolev spaces, Academic Press, Elsevier, Amsterdam, 2003. · Zbl 1098.46001
[25] A.Quarteroni and A.Valli, Numerical approximation of partial differential equations, vol. 23, Springer Series in Computational Mathematics, Springer‐VerlagBerlin, 1994. · Zbl 0803.65088
[26] P.Ciarlet, Linear and nonlinear functional analysis with applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. · Zbl 1293.46001
[27] J. E.Roberts and J. M.Thomas, Mixed and hybrid methods, P. G.Ciarlet (ed.) and J. L.Lions (ed.), editors, Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North‐Holland, Amsterdam, 1991. · Zbl 0875.65090
[28] F.Hecht, New development in FreeFem++, J Numer Math20 (2012), 251-265. · Zbl 1266.68090
[29] T.Davis, Algorithm 832: UMFPACK V4.3 ‐ an unsymmetric‐pattern multifrontal method, ACM Trans Math Softw30 (2004), 196-199. · Zbl 1072.65037
[30] L. I. G.Kovasznay, Laminar flow behind a two‐dimensional grid, Proc Camb Philos Soc44 (1948), 58-62. · Zbl 0030.22902
[31] P.Ciarlet, The finite element method for elliptic problems, North‐Holland, Amsterdam, New York, Oxford, 1978. · Zbl 0383.65058
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