Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

Babuška I, Oden JT (2004) Verification and validation in computational engineering and science: basic concepts. Comp Meths Appl Mech Eng 193: 4057–4066 Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Pure and applied mathematics. Wiley-Interscience, New York Wiberg NE, Díez P (2006) Adaptive modeling and simulation. Comp Meths Appl Mech Eng 195(4–6): 205–480 Stein E, Stephan EP (2007) Error controlled hp-adaptive FE and FE-BE methods for variational equalities and inequalities including model adaptivity. Comp Mech 39(5): 555–680 Ladevèze P, Pelle JP (2005) Mastering calculations in linear and nonlinear mechanics. Mechanical engineering. Springer, Heidelberg Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Meth Eng 24: 337–357 Babuška I, Rheinboldt WC (1978) Error estimates for adaptive finite element computations. SIAM J Numer Anal 15(4): 736–755 Babuška I, Rheinboldt WC (1978) A posteriori error estimates for the finite element method. Int J Numer Meth Eng 12(10): 1597–1615 Ladevèze P, Leguillon D (1983) Error estimate procedure in the finite element method and applications. SIAM J Numer Anal 20(3): 485–509 Demkowicz L, Oden JT, Strouboulis T (1984) Adaptative finite elements for flow problems with moving boundaries. Part I: variational principles and a posteriori error estimates. Comp Meths Appl Mech Eng 46(2): 217–251 Zhu JZ (1997) A posteriori error estimation, the relationship between different procedures. Comp Meths Appl Mech Eng 150(1–4): 411–422 Choi HW, Paraschivoiu M (2004) Adaptive computations of a posteriori finite element output bounds: a comparison of the ‘hybrid-flux’ approach and the ‘flux-free’ approach. Comp Meths Appl Mech Eng 193(36–38): 4001–4033 Parés N, Díez P, Huerta A (2006) Subdomain-based flux-free a posteriori error estimators. Comp Meths Appl Mech Eng 195(4–6): 297–323 Carstensen C, Funken SA (1999-2000) Fully reliable localized error control in the FEM. SIAM J Sci Comput 21(4): 1465–1484 Machiels L, Maday Y, Patera AT (2000) A ‘flux-free’ nodal Neumann subproblem approach to output bounds for partial differential equations. Comptes-Rendus de l’Académie des Sci Ser I Math 330(3): 249–254 Morin P, Nochetto RH, Siebert KG (2003) Local problems on stars: a posteriori error estimators, convergence, and performance. Math Comp 72: 1067–1097 Prudhomme S, Nobile F, Chamoin L, Oden JT (2004) Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. Numer Meths Partial Diff Eqs 20(2): 165–192 Moitinho de Almeida JP, Maunder EAW (2000) Recovery of equilibrium on star patches using a partition of unity technique. Int J Numer Meth Eng (accepted). doi:10.1002/nme.2623 Parés N (2005) Error assessment for functional outputs of PDE’s: bounds and goal-oriented adaptivity. Ph.D. thesis, Universitat Politècnica de Catalunya, Barcelona, Spain Sauer-Budge AM, Bonet J, Huerta A, Peraire J (2004) Computing bounds for linear functionals of exact weak solutions to Poisson’s equation. SIAM J Numer Anal 42(4): 1610–1630 Parés N, Bonet J, Huerta A, Peraire J (2006) The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations. Comp Meths Appl Mech Eng 195(4–6): 406–429 Parés N, Díez P, Huerta A (2008) Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous galerkin time discretization. Comp Meths Appl Mech Eng 197(19–20): 1641–1660 Parés N, Díez P, Huerta A (2008) Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution. Comp Meths Appl Mech Eng 197(19–20): 1661–1679 Rannacher R, Suttmeier FT (1997) A feed-back approach to error control in finite element methods: application to linear elasticity. Comp Mech 19(5): 434–446 Prudhomme S, Oden JT (1999) On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comp Meths Appl Mech Eng 176(1–4): 313–331 Sarrate J, Peraire J, Patera AT (1999) Finite element error bounds for nonlinear outputs of the Helmholtz equation. Int J Numer Meth Fluids 11(1): 17–36 Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comp Maths Appl 41: 735–756 Díez P, Calderón G (2007) Goal-oriented error estimation for transient parabolic problems. Comp Mech 39(5): 631–646 Rüter M, Stein E (2007) On the duality of finite element discretization error control in computational newtonian and eshelbian mechanics. Comp Mech 39(5): 609–630 Cast3m. http://www-cast3m.cea.fr. Last accessed: 8 August (2008) Matlab. http://www.mathworks.com/. Last accessed: 8 August (2008) Peraire J, Patera AT (1997) Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement. In: Ladevèze P, Oden JT (eds) Workshop on new advances in adaptive computational methods in mechanics. Elsevier Science Ltd, Oxford, pp 199–216 Paraschivoiu M, Peraire J, Patera AT (1997) A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations. Comp Meths Appl Mech Eng 150(1–4): 23–50 Ladevèze P (2007) Strict upper error bounds on computed outputs of interest in computational structural mechanics. Comp Mech 42(2): 271–286 Chamoin L, Ladevèze P (2007) Bounds on history-dependent or independent local quantities in viscoelasticity problems solved by approximate methods. Int J Numer Meth Eng 71: 1387–1411