A new stabilization technique for the nonconforming Crouzeix-Raviart element applied to linear elasticity

Abstract We present a stabilized finite element method for a mixed formulation of elasticity equations using the lowest order Crouzeix–Raviart element. The stabilization is done to satisfy Korn’s inequality and is based on an oblique projection operator.

[1]  A. Galántai Projectors and Projection Methods , 2003 .

[2]  Susanne C. Brenner,et al.  Korn's inequalities for piecewise H1 vector fields , 2003, Math. Comput..

[3]  Douglas N. Arnold,et al.  A Family of Discontinuous Galerkin Finite Elements for the Reissner–Mindlin Plate , 2005, J. Sci. Comput..

[4]  D. Braess Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics , 1995 .

[5]  P. Hansbo,et al.  Stabilized Crouzeix‐Raviart element for the Darcy‐Stokes problem , 2005 .

[6]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[7]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[8]  Susanne C. Brenner A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity , 1994 .

[9]  Zhangxin Chen,et al.  A new local stabilized nonconforming finite element method for the Stokes equations , 2008, Computing.

[10]  R. S. Falk Nonconforming finite element methods for the equations of linear elasticity , 1991 .

[11]  Dongwoo Sheen,et al.  A Locking-Free Nonconforming Finite Element Method for Planar Linear Elasticity , 2003, Adv. Comput. Math..

[12]  M. Feng,et al.  Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem , 2010 .

[13]  Panayot S. Vassilevski,et al.  Multiplier Spaces for the Mortar Finite Element Method in Three Dimensions , 2001, SIAM J. Numer. Anal..

[14]  E. Stephan,et al.  A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems , 2012 .

[15]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[16]  S. C. Brenner,et al.  A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity , 1993 .

[17]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[18]  Shipeng Mao,et al.  New error estimates of nonconforming mixed finite element methods for the Stokes problem , 2014 .

[19]  P. Hansbo,et al.  Discontinuous Galerkin and the Crouzeix–Raviart element : application to elasticity , 2003 .

[20]  Douglas N. Arnold,et al.  Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model , 1997 .

[21]  H. Rui,et al.  A unified stabilized mixed finite element method for coupling Stokes and Darcy flows , 2009 .

[22]  D. Braess Stability of saddle point problems with penalty , 1996 .

[23]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[24]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[25]  Daniel B. Szyld,et al.  The many proofs of an identity on the norm of oblique projections , 2006, Numerical Algorithms.

[26]  Leszek Marcinkowski,et al.  A FETI-DP Method for Crouzeix-Raviart Finite Element Discretizations , 2012, Comput. Methods Appl. Math..

[27]  Carlo Lovadina,et al.  A Low-order Nonconforming Finite Element for Reissner-Mindlin Plates , 2004, SIAM J. Numer. Anal..

[28]  Bishnu P. Lamichhane,et al.  Higher order mortar finite elements with dual Lagrange multiplier spaces and applications , 2006 .

[29]  R. Kouhia,et al.  A linear nonconforming finite element method for nearly incompressible elasticity and stokes flow , 1995 .

[30]  Xiaoyuan Yang,et al.  A nonconforming rectangular finite element pair for the Darcy–Stokes–Brinkman model , 2013 .

[31]  Mingchao Cai,et al.  A Mixed and Nonconforming FEM with Nonmatching Meshes for a Coupled Stokes-Darcy Model , 2012, J. Sci. Comput..

[32]  C. Chinosi,et al.  Nonconforming locking-free flnite elements for Reissner-Mindlin plates , 2006 .

[33]  S. C. Brenner,et al.  Linear finite element methods for planar linear elasticity , 1992 .