Computational Mechanics

, Volume 52, Issue 6, pp 1395–1415 | Cite as

Subdomain-based error techniques for generalized finite element approximations of problems with singular stress fields

  • Felício B. BarrosEmail author
  • Clovis S. de Barcellos
  • C. Armando Duarte
  • Diego A. F. Torres
Original Paper


This paper considers four types of error measures, each tailored to the generalized finite element method. Particular attention is given to two-dimensional elasticity problems with singular stress fields. The first error measure is obtained using the equilibrated element residual method. The other three estimators overcome the necessity of equilibrating the residue by employing a subdomain strategy. In this strategy, the partition of unity (PoU) property is used to decompose the error problem into local contributions over each patch of elements. The residual functional of the error problem is the same for the subdomain estimators, but the bi-linear form is different for each one of them. In the second estimator, the bi-linear form is weighted by the PoU functions associated with the patch over which the error problem is stated. No weighting appears in the bi-linear form of the third estimator. The fourth measure is proposed as an alternative strategy, in which the products of the PoU functions and test functions are introduced as weights in the weighted integral statement of the differential equation describing the error problem. The linear form of the local error problem is then identical to that of the other subdomain techniques, while the bi-linear form is stated differently, with the PoU functions directly multiplying the test functions. The goal of this study is to investigate the performance of the four estimators in two-dimensional elasticity problems with geometries that produce singularities in the stress field and concentration of the error in the numerical solution.


Error estimator Generalized finite element method  Extended finite element method Two-dimensional singular elasticity singular fields Subdomain-based residual error estimators 



The first and second authors gratefully acknowledge the support of the Brazilian research agencies CNPq (in Portuguese—Conselho Nacional de Desenvolvimento Científico e Tecnológico—Grants 305626/2010-8 and 303315/2009-1), FAPEMIG (in Portuguese “Fundação de Amparo à Pesquisa do Estado de Minas Gerais”—Grant PPM-00244-11) and the Capes Foundation (Grant BEX9278/11-7). The third author gratefully acknowledges the support of U.S. Air Force office of Scientific Research under contract number FA9550-12-1-0379.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felício B. Barros
    • 1
    Email author
  • Clovis S. de Barcellos
    • 2
  • C. Armando Duarte
    • 3
  • Diego A. F. Torres
    • 2
  1. 1.Department of Structural EngineeringFederal University of Minas Gerais Belo HorizonteBrazil
  2. 2.Graduate Program in Mechanical EngineeringFederal University of Santa Catarina FlorianópolisBrazil
  3. 3.2122 Newmark Civil Engineering LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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