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

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

doi:10.3934/nhm.2020010

Article Contents

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

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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

Early access: April 2020

Published: June 2020

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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Abstract

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

Keywords:

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

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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

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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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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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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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