Analysis of a Staggered Discontinuous Galerkin Method for Linear Elasticity

We develop a staggered discontinuous Galerkin method for linear elasticity problems and prove its a priori error estimates. In our variational formulation the symmetry of the stress tensor is imposed weakly via Lagrange multipliers but the resulting numerical stress tensor is strongly symmetric. Optimal a priori error estimates are obtained and the error estimates are robust in nearly incompressible materials. Numerical experiments illustrating our theoretical analysis are included.

[1]  M. Fortin,et al.  Dual hybrid methods for the elasticity and the Stokes problems: a unified approach , 1997 .

[2]  Michael Vogelius,et al.  Conforming finite element methods for incompressible and nearly incompressible continua , 1984 .

[3]  Douglas N. Arnold,et al.  Mixed finite elements for elasticity , 2002, Numerische Mathematik.

[4]  B. D. Veubeke.  Stress function approach , 1975 .

[5]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[6]  I. Babuska,et al.  Locking effects in the finite element approximation of elasticity problems , 1992 .

[7]  Bernardo Cockburn,et al.  A new elasticity element made for enforcing weak stress symmetry , 2010, Math. Comput..

[8]  T. Valent,et al.  Boundary Value Problems of Finite Elasticity , 1988 .

[9]  Bernardo Cockburn,et al.  Superconvergent HDG methods for linear elasticity with weakly symmetric stresses , 2013 .

[10]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[11]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[12]  Claes Johnson,et al.  Some equilibrium finite element methods for two-dimensional elasticity problems , 1978 .

[13]  Rolf Stenberg.  Weakly Symmetric Mixed Finite Elements for Linear Elasticity , 2013, ENUMATH.

[14]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[15]  M. E. Morley.  A family of mixed finite elements for linear elasticity , 1989 .

[16]  Douglas N. Arnold,et al.  Finite elements for symmetric tensors in three dimensions , 2008, Math. Comput..

[17]  R. Stenberg.  A family of mixed finite elements for the elasticity problem , 1988 .

[18]  C. P. Gupta,et al.  A family of higher order mixed finite element methods for plane elasticity , 1984 .

[19]  Bernardo Cockburn,et al.  A hybridizable discontinuous Galerkin method for linear elasticity , 2009 .

[20]  R. Stenberg.  Analysis of mixed finite elements methods for the Stokes problem: a unified approach , 1984 .

[21]  Eric T. Chung,et al.  Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions , 2009, SIAM J. Numer. Anal..

[22]  Kwang-Yeon Kim.  Analysis of some low‐order nonconforming mixed finite elements for linear elasticity problem , 2006 .

[23]  Eric T. Chung,et al.  A Staggered Discontinuous Galerkin Method for the Stokes System , 2013, SIAM J. Numer. Anal..

[24]  Douglas N. Arnold,et al.  Mixed finite element methods for linear elasticity with weakly imposed symmetry , 2007, Math. Comput..

[25]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .

[26]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[27]  J. Whiteman.  The Mathematics of Finite Elements and Applications. , 1983 .

[28]  Johnny Guzmán.  A Unified Analysis of Several Mixed Methods for Elasticity with Weak Stress Symmetry , 2010, J. Sci. Comput..

[29]  Jay Gopalakrishnan,et al.  A Second Elasticity Element Using the Matrix Bubble , 2012 .

[30]  Franco Brezzi Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics) , 1991 .

[31]  Constantin Bacuta,et al.  Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains , 2003 .

[32]  Michael Neilan,et al.  Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions , 2014, Numerische Mathematik.

[33]  Weifeng Qiu,et al.  Mixed hp-Finite Element Method for Linear Elasticity with Weakly Imposed Symmetry: Stability Analysis , 2011, SIAM J. Numer. Anal..

[34]  Shangyou Zhang,et al.  A new family of stable mixed finite elements for the 3D Stokes equations , 2004, Math. Comput..

[35]  M. Fortin,et al.  Reduced symmetry elements in linear elasticity , 2008 .

[36]  J. Thomas,et al.  Equilibrium finite elements for the linear elastic problem , 1979 .

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