American Institute of Mathematical Sciences

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June  2020, 15(2): 215-245. doi: 10.3934/nhm.2020010

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

Javier A. Almonacid 1, , Gabriel N. Gatica 2, , Ricardo Oyarzúa 3, and  Ricardo Ruiz-Baier 4,

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  September 2019 Revised  February 2020 Published  June 2020 Early access  April 2020

Fund Project: Funding: This research was partially supported by CONICYT-Chile through the project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, and Fondecyt project 1161325; by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; and by the Monash Mathematics Research Fund S05802-3951284

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.

Keywords: Boussinesq equations, augmented mixed-primal formulation, fixed-point theory, finite element methods, a priori error analysis.
Mathematics Subject Classification: 65N30, 65N12, 65N15, 35Q79, 80A20, 76R05.
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. Networks & Heterogeneous Media, 2020, 15 (2) : 215-245. doi: 10.3934/nhm.2020010
References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[3]

A. Allendes, G. R. Barrenechea and C. Naranjo, A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem, Comput. Methods Appl. Mech. Engrg., 340 (2018), 90-120.  doi: 10.1016/j.cma.2018.05.020.  Google Scholar

[4]

J. A. Almonacid and G. N. Gatica, A fully-mixed finite element method for the $n$-dimensional Boussinesq problem with temperature-dependent parameters, Comput. Methods Appl. Math., 20 (2020), 187-213.  doi: 10.1515/cmam-2018-0187.  Google Scholar

[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.  Google Scholar

[6]

M. S Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Softw., 3 (2015), 9-23.   Google Scholar

[7]

M. Alvarez, G. N. Gatica, B. Gómez-Vargas and R. Ruiz-Baier, New mixed finite element methods for natural convection with phase-change in porous media, J. Sci. Comput., 80 (2019), 141-174.  doi: 10.1007/s10915-019-00931-4.  Google Scholar

[8]

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., 49 (2015), 1399-1427.  doi: 10.1051/m2an/2015015.  Google Scholar

[9]

M. Alvarez, G. N. Gatica and R. Ruiz-Baier, A mixed-primal finite element approximation of a sedimentation-consolidation system, Math. Models Methods Appl. Sci., 26 (2016), 867-900.  doi: 10.1142/S0218202516500202.  Google Scholar

[10]

P. R. Amestoy, I. S. Duff and J.-Y. L'Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Engrg., 184 (2000), 501-520.  doi: 10.1016/S0045-7825(99)00242-X.  Google Scholar

[11]

C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par éléments finis, RAIRO Modél. Math. Anal. Numér., 29 (1995), 871-921.  doi: 10.1051/m2an/1995290708711.  Google Scholar

[12]

B. Blankenbach, F. Busse, U. Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, G. Jarvis, M. Koch, G. Marquart, D. Moore, P. Olson, H. Schmeling and T. Schnaubelt, A benchmark comparison for mantle convection codes, Geophys. J. Int., 98 (1989), 23-38.  doi: 10.1111/j.1365-246X.1989.tb05511.x.  Google Scholar

[13]

J. Boland and W. Layton, An analysis of the FEM for natural convection problems, Numer. Methods Partial Differential Equations, 6 (1990), 115-126.  doi: 10.1002/num.1690060202.  Google Scholar

[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.  Google Scholar

[15]

J. Camaño, G. N. Gatica, R. Oyarzúa and R. Ruiz-Baier, An augmented stress-based mixed finite element method for the Navier-Stokes equations with nonlinear viscosity, Numer. Methods Partial Differential Equations, 33 (2017), 1692-1725.  doi: 10.1002/num.22166.  Google Scholar

[16]

J. Camaño, R. Oyarzúa, R. Ruiz-Baier and G. Tierra, Error analysis of an augmented mixed method for the Navier-Stokes problem with mixed boundary conditions, IMA J. Numer. Anal., 38 (2018), 1452-1484.  doi: 10.1093/imanum/drx039.  Google Scholar

[17] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.   Google Scholar
[18]

Y. Y. Chen, B. W. Li and J. K. Zhang, Spectral collocation method for natural convection in a square porous cavity with local thermal equilibrium and non-equilibrium models, Int. J. Heat Mass Transfer, 64 (2013), 35-49.  doi: 10.1016/j.ijheatmasstransfer.2016.01.007.  Google Scholar

[19]

A. Çibik and S. Kaya, A projection-based stabilized finite element method for steady-state natural convection problem, J. Math. Anal. Appl., 381 (2011), 469-484.  doi: 10.1016/j.jmaa.2011.02.020.  Google Scholar

[20]

E. Colmenares, G. N. Gatica and R. Oyarzúa, Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem, Numer. Methods Partial Differential Equations, 32 (2016), 445-478.  doi: 10.1002/num.22001.  Google Scholar

[21]

E. Colmenares and M. Neilan, Dual-mixed finite element methods for the stationary Boussinesq problem, Comput. Math. Appl., 72 (2016), 1828-1850.  doi: 10.1016/j.camwa.2016.08.011.  Google Scholar

[22]

A. Dalal and M. K. Das, Natural convection in a rectangular cavity heated from below and uniformly cooled from the top and both sides, Numer. Heat Tr. A-Appl., 49 (2006), 301-322.  doi: 10.1080/10407780500343749.  Google Scholar

[23]

H. Dallmann and D. Arndt, Stabilized finite element methods for the Oberbeck-Boussinesq model, J. Sci. Comput., 69 (2016), 244-273.  doi: 10.1007/s10915-016-0191-z.  Google Scholar

[24]

M. Farhoul, S. Nicaise and L. Paquet, A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions, Math. Comp., 69 (2000), 965-986.  doi: 10.1090/S0025-5718-00-01186-8.  Google Scholar

[25]

M. Farhloul, S. Nicaise and L. Paquet, A refined mixed finite element method for the Boussinesq equations in polygonal domains, IMA J. Numer. Anal., 21 (2001), 525-551.  doi: 10.1093/imanum/21.2.525.  Google Scholar

[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.  Google Scholar

[27]

G. N. Gatica, An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions, Electron. Trans. Numer. Anal., 26 (2007), 421-438.  doi: 10.1080/00207177708922320.  Google Scholar

[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.  Google Scholar

[29]

G. N. Gatica, R. Oyarzúa and F.-J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp., 80 (2011), 1911-1948.  doi: 10.1090/S0025-5718-2011-02466-X.  Google Scholar

[30]

P. Z. Huang, W. Q. Li and Z. Y. Si, Several iterative schemes for the stationary natural convection equations at different Rayleigh numbers, Numer. Methods Partial Differential Equations, 31 (2015), 761-776.  doi: 10.1002/num.21915.  Google Scholar

[31]

K. Julien, S. Legg, J. McWilliams and J. Werne, Rapidly rotating turbulent Rayleigh-Bénard convection, J. Fluid Mech., 322 (1996), 243-273.  doi: 10.1017/S0022112096002789.  Google Scholar

[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.  Google Scholar

[33]

P. Mora and D. A. Yuen, Comparison of convection for Reynolds and Arrhenius temperature dependent viscosities, Fluid Mech. Res. Int., 2 (2018), 99-104.  doi: 10.15406/fmrij.2018.02.00025.  Google Scholar

[34]

R. Oyarzúa and P. Zúñiga, Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters, J. Comput. Appl. Math., 323 (2017), 71-94.  doi: 10.1016/j.cam.2017.04.009.  Google Scholar

[35]

C. E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier-Stokes/energy system with temperature-dependent viscosity. I: Analysis of the continuous problem, Internat. J. Numer. Methods Fluids, 56 (2008), 63-89.  doi: 10.1002/fld.1509.  Google Scholar

[36]

C. E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier-Stokes/energy system with temperature-dependent viscosity. II: The discrete problem and numerical experiments, Internat. J. Numer. Methods Fluids, 56 (2008), 91-114.  doi: 10.1002/fld.1572.  Google Scholar

[37]

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.  Google Scholar

[38]

J. E. Roberts and J. M. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, Handb. Numer. Anal., II, North-Holland, Amsterdam, 2 (1991), 523-639.   Google Scholar

[39]

M. Tabata and D. Tagami, Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients, Numer. Math., 100 (2005), 351-372.  doi: 10.1007/s00211-005-0589-2.  Google Scholar

[40]

C. Waluga, B. Wohlmuth and U. Rüde, Mass-corrections for the conservative coupling of flow and transport on collocated meshes, J. Comput. Phys., 305 (2016), 319-332.  doi: 10.1016/j.jcp.2015.10.044.  Google Scholar

[41]

J. Woodfield, M. Alvarez, B. Gómez-Vargas and R. Ruiz-Baier, Stability and finite element approximation of phase change models for natural convection in porous media, J. Comput. Appl. Math., 360 (2019), 117-137.  doi: 10.1016/j.cam.2019.04.003.  Google Scholar

[42]

T. Zhang and H. X. Liang, Decoupled stabilized finite element methods for the Boussinesq equations with temperature-dependent coefficients, Internat. J. Heat Mass Tr., 110 (2017), 151-165.  doi: 10.1016/j.ijheatmasstransfer.2017.03.002.  Google Scholar

[43]

A. G. Zimmerman and J. Kowalski, Simulating convection-coupled phase-change in enthalpy form with mixed finite elements, Preprint, (2019), arXiv: 1907.0441v1. Google Scholar

show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[3]

A. Allendes, G. R. Barrenechea and C. Naranjo, A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem, Comput. Methods Appl. Mech. Engrg., 340 (2018), 90-120.  doi: 10.1016/j.cma.2018.05.020.  Google Scholar

[4]

J. A. Almonacid and G. N. Gatica, A fully-mixed finite element method for the $n$-dimensional Boussinesq problem with temperature-dependent parameters, Comput. Methods Appl. Math., 20 (2020), 187-213.  doi: 10.1515/cmam-2018-0187.  Google Scholar

[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.  Google Scholar

[6]

M. S Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Softw., 3 (2015), 9-23.   Google Scholar

[7]

M. Alvarez, G. N. Gatica, B. Gómez-Vargas and R. Ruiz-Baier, New mixed finite element methods for natural convection with phase-change in porous media, J. Sci. Comput., 80 (2019), 141-174.  doi: 10.1007/s10915-019-00931-4.  Google Scholar

[8]

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., 49 (2015), 1399-1427.  doi: 10.1051/m2an/2015015.  Google Scholar

[9]

M. Alvarez, G. N. Gatica and R. Ruiz-Baier, A mixed-primal finite element approximation of a sedimentation-consolidation system, Math. Models Methods Appl. Sci., 26 (2016), 867-900.  doi: 10.1142/S0218202516500202.  Google Scholar

[10]

P. R. Amestoy, I. S. Duff and J.-Y. L'Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Engrg., 184 (2000), 501-520.  doi: 10.1016/S0045-7825(99)00242-X.  Google Scholar

[11]

C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par éléments finis, RAIRO Modél. Math. Anal. Numér., 29 (1995), 871-921.  doi: 10.1051/m2an/1995290708711.  Google Scholar

[12]

B. Blankenbach, F. Busse, U. Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, G. Jarvis, M. Koch, G. Marquart, D. Moore, P. Olson, H. Schmeling and T. Schnaubelt, A benchmark comparison for mantle convection codes, Geophys. J. Int., 98 (1989), 23-38.  doi: 10.1111/j.1365-246X.1989.tb05511.x.  Google Scholar

[13]

J. Boland and W. Layton, An analysis of the FEM for natural convection problems, Numer. Methods Partial Differential Equations, 6 (1990), 115-126.  doi: 10.1002/num.1690060202.  Google Scholar

[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.  Google Scholar

[15]

J. Camaño, G. N. Gatica, R. Oyarzúa and R. Ruiz-Baier, An augmented stress-based mixed finite element method for the Navier-Stokes equations with nonlinear viscosity, Numer. Methods Partial Differential Equations, 33 (2017), 1692-1725.  doi: 10.1002/num.22166.  Google Scholar

[16]

J. Camaño, R. Oyarzúa, R. Ruiz-Baier and G. Tierra, Error analysis of an augmented mixed method for the Navier-Stokes problem with mixed boundary conditions, IMA J. Numer. Anal., 38 (2018), 1452-1484.  doi: 10.1093/imanum/drx039.  Google Scholar

[17] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.   Google Scholar
[18]

Y. Y. Chen, B. W. Li and J. K. Zhang, Spectral collocation method for natural convection in a square porous cavity with local thermal equilibrium and non-equilibrium models, Int. J. Heat Mass Transfer, 64 (2013), 35-49.  doi: 10.1016/j.ijheatmasstransfer.2016.01.007.  Google Scholar

[19]

A. Çibik and S. Kaya, A projection-based stabilized finite element method for steady-state natural convection problem, J. Math. Anal. Appl., 381 (2011), 469-484.  doi: 10.1016/j.jmaa.2011.02.020.  Google Scholar

[20]

E. Colmenares, G. N. Gatica and R. Oyarzúa, Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem, Numer. Methods Partial Differential Equations, 32 (2016), 445-478.  doi: 10.1002/num.22001.  Google Scholar

[21]

E. Colmenares and M. Neilan, Dual-mixed finite element methods for the stationary Boussinesq problem, Comput. Math. Appl., 72 (2016), 1828-1850.  doi: 10.1016/j.camwa.2016.08.011.  Google Scholar

[22]

A. Dalal and M. K. Das, Natural convection in a rectangular cavity heated from below and uniformly cooled from the top and both sides, Numer. Heat Tr. A-Appl., 49 (2006), 301-322.  doi: 10.1080/10407780500343749.  Google Scholar

[23]

H. Dallmann and D. Arndt, Stabilized finite element methods for the Oberbeck-Boussinesq model, J. Sci. Comput., 69 (2016), 244-273.  doi: 10.1007/s10915-016-0191-z.  Google Scholar

[24]

M. Farhoul, S. Nicaise and L. Paquet, A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions, Math. Comp., 69 (2000), 965-986.  doi: 10.1090/S0025-5718-00-01186-8.  Google Scholar

[25]

M. Farhloul, S. Nicaise and L. Paquet, A refined mixed finite element method for the Boussinesq equations in polygonal domains, IMA J. Numer. Anal., 21 (2001), 525-551.  doi: 10.1093/imanum/21.2.525.  Google Scholar

[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.  Google Scholar

[27]

G. N. Gatica, An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions, Electron. Trans. Numer. Anal., 26 (2007), 421-438.  doi: 10.1080/00207177708922320.  Google Scholar

[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.  Google Scholar

[29]

G. N. Gatica, R. Oyarzúa and F.-J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp., 80 (2011), 1911-1948.  doi: 10.1090/S0025-5718-2011-02466-X.  Google Scholar

[30]

P. Z. Huang, W. Q. Li and Z. Y. Si, Several iterative schemes for the stationary natural convection equations at different Rayleigh numbers, Numer. Methods Partial Differential Equations, 31 (2015), 761-776.  doi: 10.1002/num.21915.  Google Scholar

[31]

K. Julien, S. Legg, J. McWilliams and J. Werne, Rapidly rotating turbulent Rayleigh-Bénard convection, J. Fluid Mech., 322 (1996), 243-273.  doi: 10.1017/S0022112096002789.  Google Scholar

[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.  Google Scholar

[33]

P. Mora and D. A. Yuen, Comparison of convection for Reynolds and Arrhenius temperature dependent viscosities, Fluid Mech. Res. Int., 2 (2018), 99-104.  doi: 10.15406/fmrij.2018.02.00025.  Google Scholar

[34]

R. Oyarzúa and P. Zúñiga, Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters, J. Comput. Appl. Math., 323 (2017), 71-94.  doi: 10.1016/j.cam.2017.04.009.  Google Scholar

[35]

C. E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier-Stokes/energy system with temperature-dependent viscosity. I: Analysis of the continuous problem, Internat. J. Numer. Methods Fluids, 56 (2008), 63-89.  doi: 10.1002/fld.1509.  Google Scholar

[36]

C. E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier-Stokes/energy system with temperature-dependent viscosity. II: The discrete problem and numerical experiments, Internat. J. Numer. Methods Fluids, 56 (2008), 91-114.  doi: 10.1002/fld.1572.  Google Scholar

[37]

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.  Google Scholar

[38]

J. E. Roberts and J. M. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, Handb. Numer. Anal., II, North-Holland, Amsterdam, 2 (1991), 523-639.   Google Scholar

[39]

M. Tabata and D. Tagami, Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients, Numer. Math., 100 (2005), 351-372.  doi: 10.1007/s00211-005-0589-2.  Google Scholar

[40]

C. Waluga, B. Wohlmuth and U. Rüde, Mass-corrections for the conservative coupling of flow and transport on collocated meshes, J. Comput. Phys., 305 (2016), 319-332.  doi: 10.1016/j.jcp.2015.10.044.  Google Scholar

[41]

J. Woodfield, M. Alvarez, B. Gómez-Vargas and R. Ruiz-Baier, Stability and finite element approximation of phase change models for natural convection in porous media, J. Comput. Appl. Math., 360 (2019), 117-137.  doi: 10.1016/j.cam.2019.04.003.  Google Scholar

[42]

T. Zhang and H. X. Liang, Decoupled stabilized finite element methods for the Boussinesq equations with temperature-dependent coefficients, Internat. J. Heat Mass Tr., 110 (2017), 151-165.  doi: 10.1016/j.ijheatmasstransfer.2017.03.002.  Google Scholar

[43]

A. G. Zimmerman and J. Kowalski, Simulating convection-coupled phase-change in enthalpy form with mixed finite elements, Preprint, (2019), arXiv: 1907.0441v1. Google Scholar

Figure 5.1.  Numerical results for Example 1. From top-left to right-bottom: XX, XY and YY components of the pseudostress, XX component of the strain rate, velocity components and vector fields, postprocessed pressure, postprocessed vorticity magnitude, and temperature. Snapshots obtained from a simulation with 214,788 DOF and a first order approximation
Figure Options

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Figure 5.2.  Numerical results for Example 2. From top-left to right-bottom: XX, XY and YY components of the pseudostress, XX component of the strain rate, velocity components and vector fields, postprocessed pressure, postprocessed vorticity magnitude, and temperature. Snapshots obtained from a simulation with 724,448 DOF using a second-order approximation
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Figure 5.3.  Example 3. Approximate solutions (from left to right and from up to down): magnitude of strain rate, pseudostress, velocity magnitude and arrows, postprocessed vorticity magnitude, postprocessed pressure, and temperature. Snapshots obtained from a simulation with a lowest-order approximation and 451,690 DOF
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Figure 4.  Example 4. Approximate velocity line integral contours and temperature profiles for the differentially heated cavity at times $ t = 4 $, $ t = 8 $, $ t = 12 $, computed with the lowest-order scheme and a backward Euler time stepping
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Table 1.  Convergence history for Example 1, with a quasi-uniform mesh refinement and approximations of first and second order
Finite Element: $ \mathbb{P}_0 $ - $ \mathbb{RT}_0 $ - $ {\bf P}_1 $ - $ P_1 $ - $ P_0 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
84 1.4140 5.2972 12.870 9.6113 1.4554 2.8012 0.8379 1.5082
268 0.7071 2.4345 7.0572 4.6912 1.0387 2.2743 0.8278 0.8069
948 0.3536 1.2700 3.8456 2.4815 0.5934 1.2154 0.3977 0.4969
3,556 0.1768 0.6461 1.9470 1.2414 0.3021 0.6162 0.2310 0.2353
13,764 0.0884 0.3248 0.9766 0.6182 0.1502 0.3084 0.0948 0.0703
54,148 0.0442 0.1626 0.4887 0.3086 0.0749 0.1542 0.0465 0.0199
214,788 0.0221 0.0814 0.2444 0.1542 0.0375 0.0771 0.0232 0.0091
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
11 0.5000 – – – – – – –
7 0.2500 1.122 0.8665 1.0352 0.4854 0.3012 0.0176 0.8069
9 0.1250 0.9385 0.8762 0.9189 0.8072 0.9037 1.0583 1.0917
8 0.0625 0.9751 0.9814 0.9989 0.9739 0.9798 0.7834 1.1912
9 0.0312 0.9924 0.9957 1.0061 1.0080 0.9988 1.2842 1.2129
8 0.0156 0.9978 0.9989 1.0020 1.0031 0.9998 1.0271 1.2816
8 0.0078 0.9994 0.9997 1.0010 1.0010 1.0000 1.0020 1.1434
Finite Element: $ \mathbb{P}_1 $ - $ \mathbb{RT}_1 $ - $ {\bf P}_2 $ - $ P_2 $ - $ P_1 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
236 1.4140 1.8442 3.3631 3.4822 0.9773 2.4131 0.7265 2.1308
820 0.7071 0.4471 1.0930 0.9907 0.2911 0.6832 0.1611 0.3965
3,044 0.3536 0.1252 0.2853 0.2855 0.0805 0.1857 0.0399 0.0833
11,716 0.1768 0.0328 0.0732 0.0747 0.0209 0.05792 0.0078 0.0213
45,956 0.0884 0.0083 0.0185 0.0189 0.0053 0.01905 0.0019 0.0056
182,020 0.0442 0.0021 0.0046 0.0047 0.0013 0.0054 0.0005 0.0011
724,448 0.0221 0.0006 0.0012 0.0012 0.0004 0.0013 0.0001 0.0002
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
6 0.5000 – – – – – – –
7 0.2500 2.0445 1.6220 1.8130 1.7471 1.6275 2.1722 2.2383
8 0.1250 1.8378 1.9377 1.7953 1.8544 1.9316 2.0114 2.2469
8 0.0625 1.9284 1.9619 1.9332 1.9440 1.9827 2.3410 1.9744
8 0.0312 1.9771 1.9871 1.9845 1.9821 1.9957 2.0073 2.0656
8 0.0156 1.9898 1.9955 1.9926 1.9932 1.9989 1.9987 2.1131
8 0.0078 1.9956 1.9970 1.9931 1.9995 1.9997 2.0031 2.2573
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Finite Element: $ \mathbb{P}_0 $ - $ \mathbb{RT}_0 $ - $ {\bf P}_1 $ - $ P_1 $ - $ P_0 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
84 1.4140 5.2972 12.870 9.6113 1.4554 2.8012 0.8379 1.5082
268 0.7071 2.4345 7.0572 4.6912 1.0387 2.2743 0.8278 0.8069
948 0.3536 1.2700 3.8456 2.4815 0.5934 1.2154 0.3977 0.4969
3,556 0.1768 0.6461 1.9470 1.2414 0.3021 0.6162 0.2310 0.2353
13,764 0.0884 0.3248 0.9766 0.6182 0.1502 0.3084 0.0948 0.0703
54,148 0.0442 0.1626 0.4887 0.3086 0.0749 0.1542 0.0465 0.0199
214,788 0.0221 0.0814 0.2444 0.1542 0.0375 0.0771 0.0232 0.0091
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
11 0.5000 – – – – – – –
7 0.2500 1.122 0.8665 1.0352 0.4854 0.3012 0.0176 0.8069
9 0.1250 0.9385 0.8762 0.9189 0.8072 0.9037 1.0583 1.0917
8 0.0625 0.9751 0.9814 0.9989 0.9739 0.9798 0.7834 1.1912
9 0.0312 0.9924 0.9957 1.0061 1.0080 0.9988 1.2842 1.2129
8 0.0156 0.9978 0.9989 1.0020 1.0031 0.9998 1.0271 1.2816
8 0.0078 0.9994 0.9997 1.0010 1.0010 1.0000 1.0020 1.1434
Finite Element: $ \mathbb{P}_1 $ - $ \mathbb{RT}_1 $ - $ {\bf P}_2 $ - $ P_2 $ - $ P_1 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
236 1.4140 1.8442 3.3631 3.4822 0.9773 2.4131 0.7265 2.1308
820 0.7071 0.4471 1.0930 0.9907 0.2911 0.6832 0.1611 0.3965
3,044 0.3536 0.1252 0.2853 0.2855 0.0805 0.1857 0.0399 0.0833
11,716 0.1768 0.0328 0.0732 0.0747 0.0209 0.05792 0.0078 0.0213
45,956 0.0884 0.0083 0.0185 0.0189 0.0053 0.01905 0.0019 0.0056
182,020 0.0442 0.0021 0.0046 0.0047 0.0013 0.0054 0.0005 0.0011
724,448 0.0221 0.0006 0.0012 0.0012 0.0004 0.0013 0.0001 0.0002
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
6 0.5000 – – – – – – –
7 0.2500 2.0445 1.6220 1.8130 1.7471 1.6275 2.1722 2.2383
8 0.1250 1.8378 1.9377 1.7953 1.8544 1.9316 2.0114 2.2469
8 0.0625 1.9284 1.9619 1.9332 1.9440 1.9827 2.3410 1.9744
8 0.0312 1.9771 1.9871 1.9845 1.9821 1.9957 2.0073 2.0656
8 0.0156 1.9898 1.9955 1.9926 1.9932 1.9989 1.9987 2.1131
8 0.0078 1.9956 1.9970 1.9931 1.9995 1.9997 2.0031 2.2573
Table 2.  Convergence history for Example 2, with a quasi-uniform mesh refinement and approximations of first and second order
Finite Element: $ \mathbb{P}_0 $ - $ \mathbb{RT}_0 $ - $ {\bf P}_1 $ - $ P_1 $ - $ P_0 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
84 0.7071 4.1107 59.150 4.6740 2.1232 3.3978 8.4722 31.684
268 0.3536 2.9724 48.185 4.8101 1.3070 2.8141 6.1465 17.922
948 0.1768 1.8371 39.145 4.9700 1.0967 2.8205 17.313 53.771
3,556 0.0884 1.5104 26.112 2.6239 0.6233 1.6445 2.6532 5.7624
13,764 0.0442 0.7732 14.525 1.283 0.3384 0.8152 1.2225 2.4021
54,148 0.0221 0.3889 7.4319 0.6359 0.1707 0.4079 0.5772 1.0130
214,788 0.0110 0.1948 3.7392 0.3178 0.0848 0.2041 0.2942 0.5675
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
7 0.5000 – – – – – – –
9 0.2500 0.4959 0.5161 0.4126 0.5271 0.2714 0.5463 0.8221
7 0.1250 0.8588 0.7512 0.7433 0.7982 0.5283 0.7894 0.8585
7 0.0625 0.9184 0.9209 0.9214 0.8147 0.7781 1.0706 1.0223
6 0.0312 0.9652 0.9438 1.0327 0.8928 1.0121 1.1193 1.2162
6 0.0156 0.9912 0.9682 1.0122 0.9854 0.9989 1.0820 1.2245
6 0.0078 0.9941 0.9778 1.0041 0.9985 0.9991 1.0357 1.1123
Finite Element: $ \mathbb{P}_1 $ - $ \mathbb{RT}_1 $ - $ {\bf P}_2 $ - $ P_2 $ - $ P_1 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
236 0.7071 3.1423 39.183 4.5951 2.1095 2.8495 7.5735 17.122
820 0.3536 1.8331 25.442 2.9427 1.5544 1.8810 3.3130 6.5814
3,044 0.1768 0.6816 10.937 1.3226 0.4762 0.8158 2.4591 3.7876
11,716 0.0884 0.2082 3.7916 0.3655 0.1364 0.1943 0.3188 0.5948
45,956 0.0442 0.0531 1.0029 0.0996 0.0322 0.0505 0.0689 0.0355
182,020 0.0221 0.0136 0.2582 0.0258 0.0082 0.0131 0.0177 0.0052
724,484 0.0110 0.0034 0.0651 0.0065 0.0021 0.0033 0.0045 0.0011
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
7 0.5000 – – – – – – –
7 0.2500 0.7249 0.4977 0.5891 0.4418 0.5991 1.1924 1.6379
6 0.1250 1.4279 1.2188 1.1543 1.7063 1.2054 0.4302 1.4443
6 0.0625 1.7114 1.5283 1.8547 1.8037 2.0781 2.9407 2.1704
6 0.0312 1.9693 1.9193 1.8755 2.0841 1.9458 2.2135 2.2071
6 0.0156 1.9614 1.9565 1.9479 1.9752 1.9473 1.9564 2.1078
6 0.0078 1.9798 1.9884 1.9810 1.9869 1.9776 1.9752 2.1907
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Finite Element: $ \mathbb{P}_0 $ - $ \mathbb{RT}_0 $ - $ {\bf P}_1 $ - $ P_1 $ - $ P_0 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
84 0.7071 4.1107 59.150 4.6740 2.1232 3.3978 8.4722 31.684
268 0.3536 2.9724 48.185 4.8101 1.3070 2.8141 6.1465 17.922
948 0.1768 1.8371 39.145 4.9700 1.0967 2.8205 17.313 53.771
3,556 0.0884 1.5104 26.112 2.6239 0.6233 1.6445 2.6532 5.7624
13,764 0.0442 0.7732 14.525 1.283 0.3384 0.8152 1.2225 2.4021
54,148 0.0221 0.3889 7.4319 0.6359 0.1707 0.4079 0.5772 1.0130
214,788 0.0110 0.1948 3.7392 0.3178 0.0848 0.2041 0.2942 0.5675
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
7 0.5000 – – – – – – –
9 0.2500 0.4959 0.5161 0.4126 0.5271 0.2714 0.5463 0.8221
7 0.1250 0.8588 0.7512 0.7433 0.7982 0.5283 0.7894 0.8585
7 0.0625 0.9184 0.9209 0.9214 0.8147 0.7781 1.0706 1.0223
6 0.0312 0.9652 0.9438 1.0327 0.8928 1.0121 1.1193 1.2162
6 0.0156 0.9912 0.9682 1.0122 0.9854 0.9989 1.0820 1.2245
6 0.0078 0.9941 0.9778 1.0041 0.9985 0.9991 1.0357 1.1123
Finite Element: $ \mathbb{P}_1 $ - $ \mathbb{RT}_1 $ - $ {\bf P}_2 $ - $ P_2 $ - $ P_1 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
236 0.7071 3.1423 39.183 4.5951 2.1095 2.8495 7.5735 17.122
820 0.3536 1.8331 25.442 2.9427 1.5544 1.8810 3.3130 6.5814
3,044 0.1768 0.6816 10.937 1.3226 0.4762 0.8158 2.4591 3.7876
11,716 0.0884 0.2082 3.7916 0.3655 0.1364 0.1943 0.3188 0.5948
45,956 0.0442 0.0531 1.0029 0.0996 0.0322 0.0505 0.0689 0.0355
182,020 0.0221 0.0136 0.2582 0.0258 0.0082 0.0131 0.0177 0.0052
724,484 0.0110 0.0034 0.0651 0.0065 0.0021 0.0033 0.0045 0.0011
IT $ \widetilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
7 0.5000 – – – – – – –
7 0.2500 0.7249 0.4977 0.5891 0.4418 0.5991 1.1924 1.6379
6 0.1250 1.4279 1.2188 1.1543 1.7063 1.2054 0.4302 1.4443
6 0.0625 1.7114 1.5283 1.8547 1.8037 2.0781 2.9407 2.1704
6 0.0312 1.9693 1.9193 1.8755 2.0841 1.9458 2.2135 2.2071
6 0.0156 1.9614 1.9565 1.9479 1.9752 1.9473 1.9564 2.1078
6 0.0078 1.9798 1.9884 1.9810 1.9869 1.9776 1.9752 2.1907
Table 3.  Convergence history for Example 3, with a quasi-uniform mesh refinement and approximations of first and second order
Finite Element: $ \mathbb{P}_0 $ - $ \mathbb{RT}_0 $ - $ {\bf P}_1 $ - $ P_1 $ - $ P_0 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
900 0.7071 2.1535 6.0574 4.7925 0.6109 1.3478 1.9131 0.0137
2,848 0.4714 1.1357 4.0980 2.9703 0.3282 1.0283 1.3774 0.0065
12,564 0.2828 0.7437 2.5440 1.8929 0.2057 0.7164 0.7827 0.0027
71,068 0.1571 0.3899 1.4422 1.1277 0.1254 0.4506 0.4332 0.0011
451,690 0.0882 0.1972 0.7612 0.6351 0.0694 0.2348 0.2179 0.0006
IT $ \tilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
7 0.7071 – – – – – – –
7 0.4714 1.0245 0.9075 0.9573 1.0982 0.8846 0.8338 1.6382
8 0.2828 0.8937 0.9558 0.9879 0.9818 0.8974 1.1057 1.6072
8 0.1571 0.9152 0.9831 0.9893 0.9874 0.9509 1.0043 1.6075
8 0.0882 0.9372 0.9852 1.0505 0.9756 0.9534 0.9891 1.6258
Finite Element: $ \mathbb{P}_1 $ - $ \mathbb{RT}_1 $ - $ {\bf P}_2 $ - $ P_2 $ - $ P_1 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
3,693 0.7071 0.7084 2.5493 2.8720 0.2803 0.7668 1.0241 0.0092
11,741 0.4714 0.2268 0.8202 0.9132 0.0846 0.1949 0.3093 0.0023
51,825 0.2828 0.0603 0.2192 0.2609 0.0217 0.0625 0.0794 0.0005
286,905 0.1571 0.0169 0.0516 0.0689 0.0575 0.0164 0.0197 0.0001
1,879,712 0.0882 0.0052 0.0135 0.0186 0.0167 0.0043 0.0051 1.84e-5
IT $ \tilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
6 0.7071 – – – – – – –
7 0.4714 1.8586 1.8163 1.8545 1.8611 1.7819 1.9314 2.5877
7 0.2828 1.9004 1.8805 1.8949 1.9072 1.9384 1.8458 2.6167
8 0.1571 1.9153 1.9572 1.8973 1.9526 1.9742 1.9628 2.5709
8 0.0882 1.9457 1.9694 1.9407 1.9644 1.9866 1.9764 2.6851
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Finite Element: $ \mathbb{P}_0 $ - $ \mathbb{RT}_0 $ - $ {\bf P}_1 $ - $ P_1 $ - $ P_0 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
900 0.7071 2.1535 6.0574 4.7925 0.6109 1.3478 1.9131 0.0137
2,848 0.4714 1.1357 4.0980 2.9703 0.3282 1.0283 1.3774 0.0065
12,564 0.2828 0.7437 2.5440 1.8929 0.2057 0.7164 0.7827 0.0027
71,068 0.1571 0.3899 1.4422 1.1277 0.1254 0.4506 0.4332 0.0011
451,690 0.0882 0.1972 0.7612 0.6351 0.0694 0.2348 0.2179 0.0006
IT $ \tilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
7 0.7071 – – – – – – –
7 0.4714 1.0245 0.9075 0.9573 1.0982 0.8846 0.8338 1.6382
8 0.2828 0.8937 0.9558 0.9879 0.9818 0.8974 1.1057 1.6072
8 0.1571 0.9152 0.9831 0.9893 0.9874 0.9509 1.0043 1.6075
8 0.0882 0.9372 0.9852 1.0505 0.9756 0.9534 0.9891 1.6258
Finite Element: $ \mathbb{P}_1 $ - $ \mathbb{RT}_1 $ - $ {\bf P}_2 $ - $ P_2 $ - $ P_1 $
DOF $ h $ $ e({\bf t}) $ $ e({{\mathit{\boldsymbol{\sigma}}}}) $ $ e({\bf u}) $ $ e(p) $ $ e({{\mathit{\boldsymbol{\gamma}}}}) $ $ e({\varphi}) $ $ e(\lambda) $
3,693 0.7071 0.7084 2.5493 2.8720 0.2803 0.7668 1.0241 0.0092
11,741 0.4714 0.2268 0.8202 0.9132 0.0846 0.1949 0.3093 0.0023
51,825 0.2828 0.0603 0.2192 0.2609 0.0217 0.0625 0.0794 0.0005
286,905 0.1571 0.0169 0.0516 0.0689 0.0575 0.0164 0.0197 0.0001
1,879,712 0.0882 0.0052 0.0135 0.0186 0.0167 0.0043 0.0051 1.84e-5
IT $ \tilde{h} $ $ r({\bf t}) $ $ r({{\mathit{\boldsymbol{\sigma}}}}) $ $ r({\bf u}) $ $ r(p) $ $ r({{\mathit{\boldsymbol{\gamma}}}}) $ $ r({\varphi}) $ $ r(\lambda) $
6 0.7071 – – – – – – –
7 0.4714 1.8586 1.8163 1.8545 1.8611 1.7819 1.9314 2.5877
7 0.2828 1.9004 1.8805 1.8949 1.9072 1.9384 1.8458 2.6167
8 0.1571 1.9153 1.9572 1.8973 1.9526 1.9742 1.9628 2.5709
8 0.0882 1.9457 1.9694 1.9407 1.9644 1.9866 1.9764 2.6851
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