Constant-free explicit error estimator with sharp upper error bound property for adaptive FE analysis in elasticity and fracture

Abstract

SUMMARY The explicit residual-based error estimator originally proposed by Babuska and Miller (1987) for adaptive finite element analysis with application to problems of linear elasticity and fracture is known to be one of the most simple and inexpensive error estimators. Indeed, it provides a theoretically guaranteed upper bound on a discretization error, measured in the energy norm, and requires small (nearly negligible) post-processing computational effort. The main issue with this classical estimator, however, is that an upper error bound is explicitly computable up to an unknown multiplicative constant. In this paper, we track the source of this generic constant, revise the original derivation procedure and derive analytically four pre-computable constants that constitute an upper error bound, resulting in, what we then call, the constant-free error estimator of the Babuska-Miller type. The performance of this constant-free estimator, as well as an adaptive FEM based on it, are illustrated on regular and singular benchmark problems and on numerical examples featuring crack propagation. The special property of the estimator is a sharp upper bound of the error: effectivity indices in the range of 1.2–2.0 are obtained, what is treated as practically (very) acceptable. In terms of efficiency judged against simplicity and inexpensiveness, the proposed constant-free explicit estimator is superior to, for example, corresponding implicit residual estimators and may be seen as a very competitive one in comparison with other known, yet more intricate and complex error estimation techniques. Copyright © 2014 John Wiley & Sons, Ltd.