AN A POSTERIORI ERROR ESTIMATOR FOR THE LAMÉ EQUATION BASED ON H ( DIV )-CONFORMING STRESS APPROXIMATIONS

Abstract. We derive a new a posteriori error estimator for the Lamé system based on H(div)conforming elements and equilibrated fluxes. It is shown that the estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established using Argyris elements. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.

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

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

[3]  Ivo Babuška,et al.  Reliable and Robust A Posteriori Error Estimation for Singularly Perturbed Reaction-Diffusion Problems , 1999 .

[4]  D. W. Kelly,et al.  Procedures for residual equilibration and local error estimation in the finite element method , 1989 .

[5]  Pierre Ladevèze,et al.  New advances on a posteriori error on constitutive relation in f.e. analysis , 1997 .

[6]  Sheng-hong Chen,et al.  Adaptive Techniques in the Finite Element Method , 2018, Springer Tracts in Civil Engineering.

[7]  Erwin Stein,et al.  Equilibrium method for postprocessing and error estimation in the finite element method , 2006 .

[8]  Adaptive Finite Element Discretization in Elasticity and Elastoplasticity by Global and Lokal Error Estimators using Local Neumann‐Problems , 1999 .

[9]  Barbara I. Wohlmuth,et al.  A Local A Posteriori Error Estimator Based on Equilibrated Fluxes , 2004, SIAM J. Numer. Anal..

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

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

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

[13]  D. Kelly,et al.  The self‐equilibration of residuals and complementary a posteriori error estimates in the finite element method , 1984 .

[14]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[15]  S. Ohnimus,et al.  Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems , 2001 .

[16]  Carsten Carstensen,et al.  A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations , 2005, Numerische Mathematik.

[17]  Yanqiu Wang,et al.  Preconditioning for the mixed formulation of linear plane elasticity , 2005 .

[18]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[19]  D. Arnold,et al.  RECTANGULAR MIXED FINITE ELEMENTS FOR ELASTICITY , 2005 .

[20]  Erwin Stein,et al.  A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems , 1998 .

[21]  Mark Ainsworth,et al.  A posteriori error estimators for second order elliptic systems part 2. An optimal order process for calculating self-equilibrating fluxes , 1993 .

[22]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

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

[24]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

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