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Guaranteed computable bounds on quantities of interest in finite element computations
Author(s) -
Ainsworth Mark,
Rankin Richard
Publication year - 2012
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.3276
Subject(s) - mathematics , estimator , multiplicative function , finite element method , order (exchange) , context (archaeology) , computation , computable general equilibrium , dirichlet boundary condition , boundary (topology) , mathematical optimization , mathematical analysis , algorithm , finance , biology , economics , thermodynamics , paleontology , statistics , physics , macroeconomics
SUMMARY We develop and compare a number of alternative approaches to obtain guaranteed and fully computable bounds on the error in quantities of interest of arbitrary order finite element approximations in the context of a linear second‐order elliptic problem. In each case, the bounds are fully computable and do not involve any unknown multiplicative factors. Guaranteed computable bounds are also obtained for the case when the Dirichlet boundary conditions are non‐homogeneous. This is achieved by taking account of the error incurred by the approximation of the Dirichlet data in the functional used to approximate the quantity of interest itself, which is found to generally give better results. Numerical examples are presented to show that the resulting estimators provide tight bounds with the effectivity index tending to unity from above. Copyright © 2012 John Wiley & Sons, Ltd.