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A posteriori error estimation for finite element solutions of Helmholtz’ equation. part I: the quality of local indicators and estimators
Author(s) -
Babuška I.,
Ihlenburg F.,
Strouboulis T.,
Gangaraj S. K.
Publication year - 1997
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/(sici)1097-0207(19970930)40:18<3443::aid-nme221>3.0.co;2-1
Subject(s) - estimator , residual , a priori and a posteriori , mathematics , helmholtz equation , finite element method , helmholtz free energy , computation , mathematical optimization , statistics , algorithm , mathematical analysis , engineering , philosophy , physics , structural engineering , epistemology , quantum mechanics , boundary value problem
This paper contains a first systematic analysis of a posteriori estimation for finite element solutions of the Helmholtz equation. In this first part, it is shown that the standard a posteriori estimates, based only on local computations, severely underestimate the exact error for the classes of wave numbers and the types of meshes employed in engineering analysis. This underestimation can be explained by observing that the standard error estimators cannot detect one component of the error, the pollution error, which is very significant at high wave numbers. Here, a rigorous analysis is carried out on a one‐dimensional model problem. The analytical results for the residual estimator are illustrated and further investigated by numerical evaluation both for a residual estimator and for the ZZ‐estimator based on smoothening. In the second part, reliable a posteriori estimators of the pollution error will be constructed. © 1997 by John Wiley & Sons, Ltd.