Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

doi:10.1007/s10483-019-2487-9

Applied Mathematics and Mechanics (English Edition) ›› 2019, Vol. 40 ›› Issue (6): 889-910.doi: https://doi.org/10.1007/s10483-019-2487-9

• 论文 • 上一篇

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

Demin LIU¹, Kaitai LI²

1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China;
2. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

收稿日期:2018-08-15 修回日期:2018-12-02 出版日期:2019-06-01 发布日期:2019-06-01
通讯作者: Kaitai LI E-mail:followtime@126.com
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11461068, 11362021, and 11401511) and the Doctoral Foundation of Xinjiang Uygur Autonomous Region of China (No. BS110101)

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

Demin LIU¹, Kaitai LI²

1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China;
2. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received:2018-08-15 Revised:2018-12-02 Online:2019-06-01 Published:2019-06-01
Contact: Kaitai LI E-mail:followtime@126.com
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11461068, 11362021, and 11401511) and the Doctoral Foundation of Xinjiang Uygur Autonomous Region of China (No. BS110101)

摘要/Abstract

摘要： In this work, the two-dimensional convective Brinkman-Forchheimer equations are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.

关键词: generalized variational principle, nonlinear structures, statically indeterminate structures, Newton iteration, Brinkman-Forchheimer equations, finite element method

Abstract: In this work, the two-dimensional convective Brinkman-Forchheimer equations are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.

Key words: generalized variational principle, nonlinear structures, statically indeterminate structures, Brinkman-Forchheimer equations, Newton iteration, finite element method

中图分类号:

65N12
65N30

Demin LIU, Kaitai LI. Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(6): 889-910.

参考文献

[1] FORCHHEIMER, P. Wasserbewegung durch Boden. Zeitschrift des Vereines Deutscher Ingeneieure, 45, 1782-1788(1901)
[2] CONSTANTIN, P. Inviscid limit for damped and driven incompressible Navier-Stokes equations in R². Communications in Mathematical Physics, 275, 529-551(2007)
[3] GEORGIEV, V. and TODOROVA, G. Existence of a solution of the wave equation with nonlinear damping and source terms. Journal of Differential Equations, 109, 295-308(1994)
[4] RAMMAHA, M. A. and STREI, T. A. Global existence and nonexistence for nonlinear wave equations with damping and source terms. Transactions of the American Mathematical Society, 354(9), 3621-3637(2002)
[5] ZHAO, C. D. and YOU, Y. C. Approximation of the incompressible convective BrinkmanForchheimer equations. Journal of Evolution Equations, 12(4), 767-788(2012)
[6] GIRAULT, V. and WHEELER, M. F. Numerical discretization of a Darcy-Forchheimer model. Numerische Mathematik, 110, 161-198(2008)
[7] PAN, H. and RUI, H. X. Mixed element method for two-dimensional Darcy-Forchheimer model. Journal of Scientific Computing, 52, 563-587(2012)
[8] ZHOU, Y. Global existence and nonexistence for a nonlinear wave equation with damping and source terms. Mathematische Nachrichten, 278(11), 1341-1358(2005)
[9] HSIAO, L. Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, Singapore (1997)
[10] ZHAO, C. S. and LI, K. T. The global attractor of Navier-Stokes equations with linear dampness on the whole two-dimensional space and estimates of its dimensions. Acta Mathematicae Applicatae Sinica, 23(1), 90-98(2000)
[11] JIANG, J. P. and HOU, Y. R. The global attractor of g-Navier-Stokes equations with linear dampness on R2. Applied Mathematics and Computation, 215, 1068-1076(2009)
[12] VAFAI, K. and TIEN, C. L. Boundary and inertia effects on flow and heat transfer in porous media. International Journal of Heat and Mass Transfer, 24(2), 195-203(1981)
[13] JOSEPH, D. D., NIELD, D. A., and PAPANICOLAOU, G. Nonlinear equation governing flow in a saturated porous medium. Water Resources Research, 18(4), 1049-1052(1982)
[14] NIELD, D. A. The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. International Journal of Heat and Mass Transfer, 12(3), 269-272(1991)
[15] NIELD, D. and BEJAN, A. Convection in Porous Media, Springer, Berlin (2006)
[16] STRAUGHAN, B. Stability and Wave Motion in Porous Media, Springer, New York (2008)
[17] SHENOY, A. Non-Newtonian fluid heat transfer in porous media. Advances in Heat Transfer, 24, 101-190(1994)
[18] UĞRULU, D. On the existence of a global attractor for the Brink man-Forchheimer equations. Nonlinear Analysis, 68, 1986-1992(2008)
[19] OUYANG, Y. and YANG, L. A note on the existence of a global attractor for the BrinkmanForchheimer equations. Nonlinear Analysis, 70, 2054-2059(2009)
[20] KALANTAROV, V. K. and ZELIK, S. Smooth attractor for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure and Applied Analysis, 11(5), 2037-2054(2012)
[21] CELEBI, A. O., KALANTAROV, V. K., and UĞURLU, D. On continuous dependence on coefficients of the Brinkman-Forchheimer equations. Applied Mathematics Letters, 19, 801-807(2006)
[22] LIU, Y. Convergence and continuous dependence for the Brinkman-Forchheimer equations. Mathematical and Computer Modelling, 49, 1401-1415(2009)
[23] LIU, D. M. and LI, K. T. Finite element analysis of the Stokes equations with damping. Mathematica Numerica Sinica, 32(4), 433-448(2010)
[24] SHI, D. Y. and YU, Z. Y. Superclose and superconvergence of finite element discretizations for the Stokes equations with damping. Applied Mathematics and Computation, 219, 7693-7698(2013)
[25] BOLING, G. and LIRENG, W. Qualitative Methods for Nonlinear Boundary Value Problems, Sun Yat-set University Press, Guangzhou (1992)
[26] ADAMS, R. A. Sobolev Spaces, Academic Press, New York (1975)
[27] GIRAULT, V. and RAVIART, P. A. Finite Elment Approximation of the Navier-Stokes Equations, Springer-Verlag, New York (1979)
[28] TEMAM, R. Navier-Stokes Equations Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam (1984)
[29] CIARLET, P. G. The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam/New York/Oxford (1978)
[30] LIU, W. B. and YAN, N. N. Some a posteriori error estimators for p-Laplacian based on residual estimation or gradient recovery. Journal of Computational Physics, 16(4), 435-477(2001)
[31] AMROUCHE, A. and GIRAULT, V. Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Mathematical Journal, 4(1), 4109-4140(1994)
[32] GHIA, U., GHIA, K. N., and SHIN, C. T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48, 387-411(1982)

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Pan WANG, Xiangcheng HAN, Weibin WEN, Baolin WANG, Jun LIANG. Galerkin-based quasi-smooth manifold element (QSME) method for anisotropic heat conduction problems in composites with complex geometry[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(1): 137-154.
[2]	Zhiping QIU, Zhao WANG, Bo ZHU. A symplectic finite element method based on Galerkin discretization for solving linear systems[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(8): 1305-1316.
[3]	Feiyang HE, Zhiyu SHI, Denghui QIAN, Y. K. LU, Yujia XIANG, Xuelei FENG. Flexural wave bandgap properties of phononic crystal beams with interval parameters[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(2): 173-188.
[4]	Chenxue JIA, Zihao WANG, Donghui ZHANG, Taihua ZHANG, Xianhong MENG. Fracture of films caused by uniaxial tensions: a numerical model[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(12): 2093-2108.
[5]	Xiaokang DU, Yuanzhao CHEN, Jing ZHANG, Xian GUO, Liang LI, Dingguo ZHANG. Nonlinear coupling modeling and dynamics analysis of rotating flexible beams with stretching deformation effect[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(1): 125-140.
[6]	Wenjie SUN, Wentao MA, Fei ZHANG, Wei HONG, Bo LI. Snap-through path in a bistable dielectric elastomer actuator[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(8): 1159-1170.
[7]	Si YUAN, Quan YUAN. Condensed Galerkin element of degree m for first-order initial-value problem with O(h^2m+2) super-convergent nodal solutions[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(4): 603-614.
[8]	S. HUSSAIN, T. TAYEBI, T. ARMAGHANI, A. M. RASHAD, H. A. NABWEY. Conjugate natural convection of non-Newtonian hybrid nanofluid in wavy-shaped enclosure[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(3): 447-466.
[9]	Xinyue LIU, Jinfang AI, Jun XIE, Guohui HU. Numerical study of opposed zero-net-mass-flow jet-induced erythrocyte mechanoporation[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(11): 1763-1776.
[10]	M. KOHANSAL-VAJARGAH, R. ANSARI, M. FARAJI-OSKOUIE, M. BAZDID-VAHDATI. Vibration analysis of two-dimensional structures using micropolar elements[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(7): 999-1012.
[11]	Ying LIU, Yanping CHEN, Yunqing HUANG, Qingfeng LI. Analysis of a two-grid method for semiconductor device problem[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(1): 143-158.
[12]	Yihui DA, Bin WANG, D. Z. LIU, Zhenghua QIAN. An analytical approach to reconstruction of axisymmetric defects in pipelines using T(0, 1) guided waves[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(10): 1479-1492.
[13]	M. SHOJAEIFARD, M. R. BAYAT, M. BAGHANI. Swelling-induced finite bending of functionally graded pH-responsive hydrogels: a semi-analytical method[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(5): 679-694.
[14]	Yang WANG, Yanping CHEN, Yunqing HUANG, Ying LIU. Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(11): 1657-1676.
[15]	M. SHEIKHOLESLAMI. Investigation of Coulomb force effects on ethylene glycol based nanofluid laminar flow in a porous enclosure[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(9): 1341-1352.