American Institute of Mathematical Sciences

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

 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.

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 and 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. [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. 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. [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. [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] 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. [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. [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. [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. [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. [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. [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. [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. [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] 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. [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. [17] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. [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. [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. [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. [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. [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. [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. [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. [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. [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] 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. [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] 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. [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. [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. [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] 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. [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. [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. [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. [37] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994. [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. [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. [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. [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. [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. [43] A. G. Zimmerman and J. Kowalski, Simulating convection-coupled phase-change in enthalpy form with mixed finite elements, Preprint, (2019), arXiv: 1907.0441v1.

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. [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. 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. [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. [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] 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. [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. [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. [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. [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. [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. [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. [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. [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] 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. [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. [17] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. [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. [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. [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. [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. [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. [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. [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. [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. [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] 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. [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] 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. [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. [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. [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] 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. [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. [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. [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. [37] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994. [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. [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. [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. [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. [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. [43] A. G. Zimmerman and J. Kowalski, Simulating convection-coupled phase-change in enthalpy form with mixed finite elements, Preprint, (2019), arXiv: 1907.0441v1.
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
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
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
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
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
 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
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
 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
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
 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
 [1] Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034 [2] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [3] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [4] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [5] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [6] Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024 [7] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014 [8] Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807 [9] Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126 [10] Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101 [11] Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure and Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339 [12] Na Peng, Jiayu Han, Jing An. An efficient finite element method and error analysis for fourth order problems in a spherical domain. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022021 [13] Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337 [14] Mei-Qin Zhan. Finite element analysis and approximations of phase-lock equations of superconductivity. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 95-108. doi: 10.3934/dcdsb.2002.2.95 [15] Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 [16] Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, 2021, 29 (5) : 3171-3191. doi: 10.3934/era.2021032 [17] Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control and Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 [18] Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092 [19] Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074 [20] Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control and Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217

2020 Impact Factor: 1.213

Metrics

• PDF downloads (283)
• HTML views (257)
• Cited by (0)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]