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

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
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  • 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.

    Mathematics Subject Classification: 65N30, 65N12, 65N15, 35Q79, 80A20, 76R05.

    Citation:

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  • 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 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

    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

    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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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