Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

In this paper we present two types of local error estimators for the primal finite-element-method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then—as a new contribution—from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post-processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore—for better numerical efficiency—the residua are projected energy-invariant onto reference elements, where the local Neumann problems have to be solved. Comparative examples between REM- and PEM-type local estimators show superior effectivity indices for the latter one. Copyright © 2001 John Wiley & Sons, Ltd.

[1]  Claes Johnson,et al.  Adaptive finite element methods in computational mechanics , 1992 .

[2]  I. Babuska,et al.  A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator , 1987 .

[3]  Serge Prudhomme,et al.  On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors , 1999 .

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

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

[6]  Zur Plattenberechnung mittels finiter Elemente , 1970 .

[7]  S. Ohnimus,et al.  Coupled model- and solution-adaptivity in the finite-element method , 1997 .

[8]  R. Rannacher,et al.  A feed-back approach to error control in finite element methods: application to linear elasticity , 1997 .

[9]  Erwin Stein,et al.  An equilibrium method for stress calculation using finite element displacement models , 1977 .

[10]  S. Ohnimus,et al.  Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems , 1999 .

[11]  Ekkehard Ramm,et al.  A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem , 1998 .

[12]  I. Babuska,et al.  Pollution Error in the h-Version of the Finite Element Method and the Local Quality of the Recovered Derivatives. , 1997 .