Symmetry-Free, p-Robust Equilibrated Error Indication for the hp-Version of the FEM in Nearly Incompressible Linear Elasticity

Abstract. We consider the extension of the p-robust equilibrated error estimator due to Braess, Pillwein and Schöberl to linear elasticity. We derive a formulation where the local mixed auxiliary problems do not require symmetry of the stresses. The resulting error estimator is p-robust, and the reliability estimate is also robust in the incompressible limit if quadratics are included in the approximation space. Extensions to other systems of linear second-order partial differential equations are discussed. Numerical simulations show only moderate deterioration of the effectivity index for a Poisson ratio close to .

[1]  M. Ainsworth,et al.  Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity , 2010 .

[2]  L. Demkowicz,et al.  Analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes , 2007 .

[3]  Marta Lewicka,et al.  The uniform Korn–Poincaré inequality in thin domains , 2008, 0803.0355.

[4]  Kwang-Yeon Kim.  A POSTERIORI ERROR ESTIMATOR FOR LINEAR ELASTICITY BASED ON NONSYMMETRIC STRESS TENSOR APPROXIMATION , 2012 .

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

[6]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[7]  Serge Nicaise,et al.  An a posteriori error estimator for the Lamé equation based on equilibrated fluxes , 2007 .

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

[9]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[10]  S. Repin.  A Posteriori Estimates for Partial Differential Equations , 2008 .

[11]  Cornelius O. Horgan,et al.  Korn's Inequalities and Their Applications in Continuum Mechanics , 1995, SIAM Rev..

[12]  Carsten Carstensen,et al.  A unifying theory of a posteriori finite element error control , 2005, Numerische Mathematik.

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

[14]  Dietrich Braess,et al.  Equilibrated residual error estimator for edge elements , 2007, Math. Comput..

[15]  Dietrich Braess,et al.  Equilibrated residual error estimates are p-robust , 2009 .

[16]  William F. Mitchell,et al.  A comparison of adaptive refinement techniques for elliptic problems , 1989, TOMS.

[17]  D. Schötzau,et al.  An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity , 2006 .

[18]  C. Carstensen,et al.  Partition of unity for localization in implicit a posteriori finite element error control for linear elasticity , 2008 .

[19]  R. Verfürth.  A review of a posteriori error estimation techniques for elasticity problems , 1999 .

[20]  Mark Ainsworth,et al.  Realistic computable error bounds for three dimensional finite element analyses in linear elasticity , 2011 .

[21]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[22]  I. Babuska,et al.  On locking and robustness in the finite element method , 1992 .

[23]  I. Babuska,et al.  Finite Element Analysis , 2021 .

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

[25]  J. Melenk,et al.  An adaptive strategy for hp-FEM based on testing for analyticity , 2007 .

[26]  L. Demkowicz,et al.  Mixed hp-Finite Element Method for Linear Elasticity with Weakly Imposed Symmetry: Stability Analysis , 2009, SIAM J. Numer. Anal..

[27]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[28]  I. Babuska,et al.  Locking effects in the finite element approximation of plate models , 1995 .

[29]  Barbara I. Wohlmuth,et al.  On residual-based a posteriori error estimation in hp-FEM , 2001, Adv. Comput. Math..

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

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

[32]  Dietrich Braess.  Finite Elements: Introduction , 2007 .

[33]  D. Arnold,et al.  Mixed Finite Elements for Elasticity in the Stress-Displacement Formulation , 2002 .

[34]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[35]  L. R. Scott,et al.  Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials , 1985 .