Quadratic nonconforming finite element method for 3D Stokes equations on cuboid meshes

详细信息    查看全文

• 作者：Xin-chen Zhou ; Zhao-liang Meng…
• 关键词：Stokes equations ; nonconforming finite element ; macroelement ; stability
• 刊名：Applied Mathematics - A Journal of Chinese Universities
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：31
• 期：1
• 页码：21-36
• 全文大小：276 KB
• 参考文献：[1]D N Arnold, F Brezzi, M Fortin. A stable finite element for the Stokes equations, Calcolo, 1984, 21: 337–344.MathSciNet CrossRef MATH
  [2]L Bai. A nonconforming mixed finite element method for 3D Stokes equations, Master thesis, Dalian University of Technology, 2013.
  [3]F Brezzi, M Fortin. Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.CrossRef MATH
  [4]M Crouzeix, P A Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations, Rev Fr Autom Inform Rech Oper Math, 1973, 3: 33–75.MathSciNet MATH
  [5]M Fortin. An analysis of the convergence of mixed finite element methods, ESAIM Math Model Numer Anal, 1977, 11: 341–354.MathSciNet MATH
  [6]V Girault, P A Raviart. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Berlin-Heidelberg-New York, Springer, 1986.CrossRef MATH
  [7]P Hood, C Taylor. Numerical solution of the Navier-Stokes equations using the finite element technique, Comput Fluids, 1973, 1: 1–28.MathSciNet CrossRef MATH
  [8]J Hu, H Man, Z Shi. Constrained nonconforming rotated Q1 element for stokes flow and planar elasticity, Math Numer Sinica, 2005, 27: 311–324.MathSciNet
  [9]G Matthies. Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra, ESAIM Math Model Numer Anal, 2007, 41: 855–874.MathSciNet CrossRef MATH
  [10]Z Meng, D Sheen, Z Luo, S Kim. Three-dimensional quadratic nonconforming brick element, Numer Methods Partial Differential Equations, 2014, 30: 158–174.MathSciNet CrossRef MATH
  [11]H Nam, H J Choi, C Park, D Sheen. A cheapest nonconforming rectangular finite element for the stationary Stokes problem, Comput Methods Appl Mech Engrg, 2013, 257: 77–86.MathSciNet CrossRef MATH
  [12]C Park, D Sheen, B C Shin. A subspace of the DSSY nonconforming quadrilateral finite element space for the Stokes equations, J Comput Appl Math, 2013, 239: 220–230.MathSciNet CrossRef MATH
  [13]R Stenberg. Analysis of mixed finite element methods for the Stokes problem: a unified approach, Math Comput, 1984, 42: 9–23.MathSciNet MATH
  [14]S Zhang. A new family of stable mixed finite elements for the 3D Stokes equations, Math Comput, 2005, 74(250): 543–554.MathSciNet CrossRef MATH
  
• 作者单位：Xin-chen Zhou (1)
  Zhao-liang Meng (1)
  Xiao-shan Wang (1)
  Zhong-xuan Luo (2)
  
  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
  2. School of Software, Dalian University of Technology, Dalian, 116024, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Applications of Mathematics
  Chinese Library of Science
  
• 出版者：Editorial Committee of Applied Mathematics - A Journal of Chinese Universities
• ISSN：1993-0445

文摘

In this paper, the quadratic nonconforming brick element (MSLK element) introduced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK−P ₁ is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P ₁ element approximates the pressure. As a cure, we adopt the piecewise P ₁ macroelement to discretize the pressure instead of the standard piecewise P ₁ element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.