带有阻尼项的定常Stokes方程的低阶非协调混合有限元方法的超逼近和超收敛分析

Abstract

Abstract:

In this paper, we apply the constrained nonconforming rotated Q₁element and the piecewise constant element to approximate the velocity and the pressure for the stationary、impressible Stokes equations with damped term, respectively. The existence and uniqueness of the approximated solutions are proved. Employing the prior estimates of the exact and approximate solutions and choosing the appropriate parameters α, ν and r, the optimal error estimates and the superclose results are derived. Finally, the O(h²) order global superconvergence in H¹-norm for the velocity and L²-norm for the pressure is obtained by use of a postprocessing technique.

Key words: Stokes equations, Damped term, Nonconforming mixed elements, Superclose and superconvergence, Optimal error estimates

CLC Number:

65N30

SHI Dong-Yang, YU Zhi-Yun. Superclose and Superconvergence Analysis of a Low Order Nonconforming Mixed Finite Element Method for Stationary
Stokes Equations with Damping[J].Acta mathematica scientia,Series A, 2013, 33(4): 735-745.

Trendmd

References

[1] Wahlbin L B. Superconvergence in Galerkin Finite Element Methods, Vol 1605. Berlin: Springer, 1995

[2] Zhu Q. A review of two different approaches for superconvergence analysis. Appl Math, 1998, 43(6): 401--411

[3] Li J, Wheeler M F. Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids. SIAM J Numer Anal, 2000, 38(3): 770--798

[4] Lin Q, Lin J. Finite Element Methods: Accuracy and Improvement. Beijing: Science Press, 2006

[5] 林群, 严宁宁. 高效有限元构造与分析. 保定:河北大学出版社, 1996

[6] Yan N. Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods. Beijing: Science Press, 2008

[7] Matthies G, Skrzypacz P, Tobiska L. Superconvergence of a 3D finite element method for stationary Stokes and
Navier-Stokes problems. Numer Meth PDEs, 2005, 21(4): 701--725

[8] Li J, Mei L, Chen Z. Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh L² projection. Numer Meth PDEs, 2010, DOI 10.1002/num.20610

[9] Eichel H, Tobiska L, Xie H. Superclosness and superconvergence of stabilized low order finite element discretizations of the Stokes problem. Math Comput, 2011, 80(274): 697--722

[10] Chen W, Chen P, Gunzburger M, Yan N. Superconvergence analysis of FEMs for the Stokes-Darcy system. Math Methods Appl Sci, 2010, 33: 1605--1617

[11] Antman S S. The equations for large vibrations of strings. Amer Math Monthly, 1980, 87: 359--370

[12] Georgiev V, Todorova G. Existence of a solution of the wave equation with nonlinear damping and source terms. J Differential Equations, 1994, 109: 295--308

[13] Bresch D, Desjardins B. Existence of global weak solutions for a 2D viscous shallow water equations and
convergence to the quasi-geostrophic model. Comm Math Phys, 2003, 238: 211--223

[14] Zhou Y. Global existence and nonexistence for a nonlinear wave equation with damping and source terms. Math Nach, 2005, 278(11): 1341--1358

[15] Constantin P. Inviscid limit for damped and driven incompressible Navier-Stokes equations in R². Comm Math Phys, 2007, 275: 529--551

[16] 丁夏畦, 吴永辉. 二维全平面上具线性阻尼Navier-Stokes方程组解的有限维行为. 应用数学学报, 1997, 20(4): 509--520

[17] 赵春山, 李开泰. 二维全空间上线性阻尼Navier-Stokes方程的全局吸引子及其维数估计. 应用数学学报, 2000, 23(1): 90--98

[18] Cai X, Jiu Q. Weak and strong solutions for the incompressible Navier-Stokes equations with damping term. J Math Anal Appl, 2008, 343: 799--809

[19] Cai X, Lei L. L² decay of the incompressible Navier-Stokes equations with damping. Acta Math Scientia, 2010, 30B(4): 1235--1248

[20] 刘德民, 李开泰. 带有阻尼项的Stokes 方程的有限元分析. 计算数学, 2010, 32(4): 433--448

[21] Lin Q, Tobiska L, Zhou A. Superconvergence and extrapolation of nonconforming low order elements applied to the
Poisson equation. IMA J Numer Anal, 2005, 25(1): 160--181

[22] Shi D Y, Liang H. Superconvergence analysis and extrapolation of a new unconventional Hermite-type anisotropic rectangular element. Math Numer Sinica, 2005, 27(4): 369--382

[23] Shi D Y, Peng Y C, Chen S C. Superconvergence of a nonconforming finite element approximation to viscoelasticity type equations on anisotropic meshes. Numer Math A Journal of Chinese Universities (English Series), 2006, 15: 375--384

[24] Qiao Z H, Yao C H, Jia S H. Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-Harmonic Maxwell's equations. J Sci Comput, 2011, 46: 1--19

[25] Wang J, Ye X. Superconvergence of finite element approximations for the Stokes problem by projection methods. SIAM J Numer Anal, 2002, 30: 1001--1013

[26] Ye X. Superconvergence of nonconforming finite element method for the Stokes equations. Numer Meth PDEs, 2002, 18: 143--154

[27] 石东洋, 王彩霞. Stokes问题非协调混合有限元超收敛分析. 应用数学学报, 2007, 30(6): 1056--1065

[28] Liu H, Yan N. Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations. Adv Comput Math, 2008, 29: 375--392

[29] 胡俊, 满红英, 石钟慈. 带约束非协调旋转Q₁ 元在Stokes和平面弹性问题的应用. 计算数学, 2005, 27(3): 311--324

[30] Rannacher R, Turek S. Simple nonconforming quadrilateral Stokes element. Numer Meth for PDEs, 1992, 8: 97--111

[31] Apel T, Nicaise S, Sch"oberl J. Crouzeix-Raviart type finite elements on anisotropic meshes. Numer Math, 2001, 89: 193--223

[32] Shi D Y, Mao S P. An anisotropic nonconforming finite element with some superconvergence resultes. J Comput Math, 2005, 23(3): 261--274

[33] Shi D Y, Ren J C. Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes. Nonlinear Anal: TMA, 2009, 71(9): 3842--3852

[34] Shi D Y, Ren J C. Nonconforming mixed finite element method for the stationary conduction-convection problem. Int J Numer Anal Model, 2009, 6(2): 293--310

[35] Shi D Y, Wang C. A new low order nonconforming mixed finite element scheme for second order elliptic
problems. Inter J Comput Math, 2011, 88(10): 2167--2177

[36] Shi D Y, Ren J, Gong W. A new nonconforming mixed finite element scheme for the stationary Navier-Stokes equations. Acta Math Sci, 2011, 31(2): 367--382

Superclose and Superconvergence Analysis of a Low Order Nonconforming Mixed Finite Element Method for Stationary
Stokes Equations with Damping

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	ZHANG Zu-Jin. An Improved Regularity Criterion for the 3D Navier-Stokes Equations in Terms of Two Entries of the Velocity Gradient [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1327-1335.
[2]	YANG Xin-Guang, WANG Hong-Jun, LI Jun-Tao. Uniform Attractors for the 2D Non-Autonomous Navier-Stokes Equation with Weak Damping [J]. Acta mathematica scientia,Series A, 2014, 34(4): 828-840.
[3]	GAO Zhen-Sheng, JIANG Fei, WANG Wei-Wei. Semi-Strong Solutions to Hydrodynamic Flow of Liquid Crystals [J]. Acta mathematica scientia,Series A, 2014, 34(2): 367-377.
[4]	DONG Jian-Wei, ZHANG You-Lin, WANG Yan-Ping. Analysis of the Stationary Quantum Navier-Stokes Equations in one Space Dimension [J]. Acta mathematica scientia,Series A, 2013, 33(4): 719-727.
[5]	SONG Hong-Li, GUO Zhen-Hua. Existence of Global Strong Solutions and Interface Behavior of Solutions for 1D Compressible Navier-Stokes Equations with Free Boundary Value Problem [J]. Acta mathematica scientia,Series A, 2013, 33(4): 601-620.
[6]	BIAN Dong-Fen, YUAN Bao-Quan. Regularity of Weak Solutions to the Generalized Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2011, 31(6): 1601-1609.
[7]	ZHAO Cai-De. H¹-uniform Attractor for 2D Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1416-1430.
[8]	YUAN Hong-Jun, WANG Shu. The Zero-Mach Limit of the Compressible Convection [J]. Acta mathematica scientia,Series A, 2011, 31(1): 53-63.
[9]	SUN Jian-Zhu, FAN Ji-Shan. Regularity Criteria for a Two-fluid Model of the Truncated Euler Equations [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1693-1698.
[10]	SHI Dong-Yang, WANG Hui-Min. The Lumped Mass Nonconforming Finite Element Approximation for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes [J]. Acta mathematica scientia,Series A, 2010, 30(4): 1018-1029.
[11]	Li Kaitai;Jia Huilian. The Navier-Stokes Equations in Stream Layer or on Stream Surface and a Dimension Split Method [J]. Acta mathematica scientia,Series A, 2008, 28(2): 266-282.
[12]	Zhang Ting. Discontinuous Solutions of the Navier-Stokes Equations for Compressible Flow with Density-Dependent Viscosity [J]. Acta mathematica scientia,Series A, 2008, 28(2): 214-221.
[13]	Dong Boqing; Li Yongsheng. Large Time Behavior of the Modified Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2006, 26(4): 498-505.
[14]	WANG He-Yuan, DING Su-Zhen, XI Chu-Wei. Numerical Simulation of Spherical Couette Flow [J]. Acta mathematica scientia,Series A, 2005, 25(2): 245-250.
[15]	He Yinnian. Optimum Nonlinear Galerkin Algorithm for the Penalized Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 1998, 18(3): 251-256.

Superclose and Superconvergence Analysis of a Low Order Nonconforming Mixed Finite Element Method for Stationary Stokes Equations with Damping

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

Superclose and Superconvergence Analysis of a Low Order Nonconforming Mixed Finite Element Method for Stationary
Stokes Equations with Damping