A new mixed finite element method for the <i>n</i>-dimensional Boussinesq problem with temperature-dependent viscosity

Javier A. Almonacid; Gabriel N. Gatica; Ricardo Oyarzúa; Ricardo Ruiz-Baier; Javier A. Almonacid; Gabriel N. Gatica; Ricardo Oyarzúa; Ricardo Ruiz-Baier

doi:10.3934/nhm.2020010

Networks and Heterogeneous Media

2020, Volume 15, Issue 2: 215-245. doi: 10.3934/nhm.2020010

Previous Article Next Article

A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity

1.
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada
2.
Centro de Investigación en Ingeniería Matemática (CI²MA); and, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
3.
GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile, and, Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Concepción, Chile
4.
School of Mathematics, Monash University, 9 Rainforest Walk, Clayton, Victoria 3800, Australia, and, Universidad Adventista de Chile, Casilla 7-D Chillán, Chile, and, Laboratory of Mathematical Modelling, Institute of Personalized Medicine, Sechenov University, Moscow, Russian Federation

Received: 01 September 2019 Revised: 01 February 2020 Published: 10 April 2020
65N30, 65N12, 65N15, 35Q79, 80A20, 76R05

In this paper we propose a new mixed-primal formulation for heat-driven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixed-point theorem and the generalized theory of saddle-point problems. The motivation is to overcome a drawback in a recent work by the authors where, in the mixed formulation for the momentum equation, the reciprocal of the viscosity is a pre-factor to a tensor product of velocities; making the analysis quite restrictive, as one needs a given continuous injection that holds only in 2D. We show in this work that by adding both the pseudo-stress and the strain rate tensors as new unknowns in the problem, we get more flexibility in the analysis, covering also the 3D case. The rest of the formulation is based on eliminating the pressure, incorporating augmented Galerkin-type terms in the mixed form of the momentum equation, and defining the normal heat flux as a suitable Lagrange multiplier in a primal formulation for the energy equation. Additionally, the symmetry of the stress is imposed in an ultra-weak sense, and consequently the vorticity tensor is no longer required as part of the unknowns. A finite element method that follows the same setting is then proposed, where we remark that both pressure and vorticity can be recovered from the principal unknowns via postprocessing formulae. The solvability of the discrete problem is analyzed by means of the Brouwer fixed-point theorem, and we derive error estimates in suitable norms. Numerical examples illustrate the performance of the new schem and its use in the simulation of mantle convection, and they also confirm the theoretical rates of convergence.
- Boussinesq equations,
- augmented mixed-primal formulation,
- fixed-point theory,
- finite element methods,
- a priori error analysis
Citation: Javier A. Almonacid, Gabriel N. Gatica, Ricardo Oyarzúa, Ricardo Ruiz-Baier. A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity[J]. Networks and Heterogeneous Media, 2020, 15(2): 215-245. doi: 10.3934/nhm.2020010

Related Papers:

Abstract

In this paper we propose a new mixed-primal formulation for heat-driven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixed-point theorem and the generalized theory of saddle-point problems. The motivation is to overcome a drawback in a recent work by the authors where, in the mixed formulation for the momentum equation, the reciprocal of the viscosity is a pre-factor to a tensor product of velocities; making the analysis quite restrictive, as one needs a given continuous injection that holds only in 2D. We show in this work that by adding both the pseudo-stress and the strain rate tensors as new unknowns in the problem, we get more flexibility in the analysis, covering also the 3D case. The rest of the formulation is based on eliminating the pressure, incorporating augmented Galerkin-type terms in the mixed form of the momentum equation, and defining the normal heat flux as a suitable Lagrange multiplier in a primal formulation for the energy equation. Additionally, the symmetry of the stress is imposed in an ultra-weak sense, and consequently the vorticity tensor is no longer required as part of the unknowns. A finite element method that follows the same setting is then proposed, where we remark that both pressure and vorticity can be recovered from the principal unknowns via postprocessing formulae. The solvability of the discrete problem is analyzed by means of the Brouwer fixed-point theorem, and we derive error estimates in suitable norms. Numerical examples illustrate the performance of the new schem and its use in the simulation of mantle convection, and they also confirm the theoretical rates of convergence.

References

[1]	(2003) Sobolev Spaces. Amsterdam: Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press.
[2]	R. Aldbaissy, F. Hecht, G. Mansour and T. Sayah, A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity, Calcolo, 55 (2018), Art. 44, 49 pp. doi: 10.1007/s10092-018-0285-0
[3]	A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem. Comput. Methods Appl. Mech. Engrg. (2018) 340: 90-120.
[4]	A fully-mixed finite element method for the $n$-dimensional Boussinesq problem with temperature-dependent parameters. Comput. Methods Appl. Math. (2020) 20: 187-213.
[5]	J. A. Almonacid, G. N. Gatica and R. Oyarzúa, A mixed-primal finite element method for the Boussinesq problem with temperature-dependent viscosity, Calcolo, 55 (2018), Art. 36, 42 pp. doi: 10.1007/s10092-018-0278-z
[6]	The FEniCS project version 1.5. Arch. Numer. Softw. (2015) 3: 9-23.
[7]	New mixed finite element methods for natural convection with phase-change in porous media. J. Sci. Comput. (2019) 80: 141-174.
[8]	An augmented mixed-primal finite element method for a coupled flow-transport problem. ESAIM Math. Model. Numer. Anal. (2015) 49: 1399-1427.
[9]	A mixed-primal finite element approximation of a sedimentation-consolidation system. Math. Models Methods Appl. Sci. (2016) 26: 867-900.
[10]	Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Engrg. (2000) 184: 501-520.
[11]	Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par éléments finis. RAIRO Modél. Math. Anal. Numér. (1995) 29: 871-921.
[12]	A benchmark comparison for mantle convection codes. Geophys. J. Int. (1989) 98: 23-38.
[13]	An analysis of the FEM for natural convection problems. Numer. Methods Partial Differential Equations (1990) 6: 115-126.
[14]	F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1
[15]	An augmented stress-based mixed finite element method for the Navier-Stokes equations with nonlinear viscosity. Numer. Methods Partial Differential Equations (2017) 33: 1692-1725.
[16]	Error analysis of an augmented mixed method for the Navier-Stokes problem with mixed boundary conditions. IMA J. Numer. Anal. (2018) 38: 1452-1484.
[17]	(1961) Hydrodynamic and Hydromagnetic Stability. Oxford: The International Series of Monographs on Physics Clarendon Press.
[18]	Spectral collocation method for natural convection in a square porous cavity with local thermal equilibrium and non-equilibrium models. Int. J. Heat Mass Transfer (2013) 64: 35-49.
[19]	A projection-based stabilized finite element method for steady-state natural convection problem. J. Math. Anal. Appl. (2011) 381: 469-484.
[20]	Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem. Numer. Methods Partial Differential Equations (2016) 32: 445-478.
[21]	Dual-mixed finite element methods for the stationary Boussinesq problem. Comput. Math. Appl. (2016) 72: 1828-1850.
[22]	Natural convection in a rectangular cavity heated from below and uniformly cooled from the top and both sides. Numer. Heat Tr. A-Appl. (2006) 49: 301-322.
[23]	Stabilized finite element methods for the Oberbeck-Boussinesq model. J. Sci. Comput. (2016) 69: 244-273.
[24]	A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions. Math. Comp. (2000) 69: 965-986.
[25]	A refined mixed finite element method for the Boussinesq equations in polygonal domains. IMA J. Numer. Anal. (2001) 21: 525-551.
[26]	E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Int. J. Diff. Eqns., 2006 (2006), Art. ID 90616, 14 pp. doi: 10.1155/denm/2006/90616
[27]	An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions. Electron. Trans. Numer. Anal. (2007) 26: 421-438.
[28]	G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, 2014. doi: 10.1007/978-3-319-03695-3
[29]	Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comp. (2011) 80: 1911-1948.
[30]	Several iterative schemes for the stationary natural convection equations at different Rayleigh numbers. Numer. Methods Partial Differential Equations (2015) 31: 761-776.
[31]	Rapidly rotating turbulent Rayleigh-Bénard convection. J. Fluid Mech. (1996) 322: 243-273.
[32]	M. Kaddiri, M. Naïmi, A. Raji and M. Hasnaoui, Rayleigh-Bénard convection of non-Newtonian power-law fluids with temperature-dependent viscosity, Int. Schol. Res. Netw., (2012), 614712. doi: 10.5402/2012/614712
[33]	Comparison of convection for Reynolds and Arrhenius temperature dependent viscosities. Fluid Mech. Res. Int. (2018) 2: 99-104.
[34]	Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters. J. Comput. Appl. Math. (2017) 323: 71-94.
[35]	The steady Navier-Stokes/energy system with temperature-dependent viscosity. I: Analysis of the continuous problem. Internat. J. Numer. Methods Fluids (2008) 56: 63-89.
[36]	The steady Navier-Stokes/energy system with temperature-dependent viscosity. II: The discrete problem and numerical experiments. Internat. J. Numer. Methods Fluids (2008) 56: 91-114.
[37]	A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.
[38]	Mixed and hybrid methods. Handbook of Numerical Analysis, Handb. Numer. Anal., II, North-Holland, Amsterdam (1991) 2: 523-639.
[39]	Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. (2005) 100: 351-372.
[40]	Mass-corrections for the conservative coupling of flow and transport on collocated meshes. J. Comput. Phys. (2016) 305: 319-332.
[41]	Stability and finite element approximation of phase change models for natural convection in porous media. J. Comput. Appl. Math. (2019) 360: 117-137.
[42]	Decoupled stabilized finite element methods for the Boussinesq equations with temperature-dependent coefficients. Internat. J. Heat Mass Tr. (2017) 110: 151-165.
[43]	A. G. Zimmerman and J. Kowalski, Simulating convection-coupled phase-change in enthalpy form with mixed finite elements, Preprint, (2019), arXiv: 1907.0441v1.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)