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IMA Journal of Numerical Analysis Advance Access originally published online on February 27, 2008
IMA Journal of Numerical Analysis 2009 29(1):126-140; doi:10.1093/imanum/drm026
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A safeguarded dual weighted residual method

Ricardo H. Nochetto{dagger}

Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

Andreas Veeser{ddagger}

Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Milano 20133, Italy

Marco Verani§

MOX—Modeling and Scientific Computing, Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, Milano 20133, Italy

{dagger} Email: rhn{at}math.umd.edu

{ddagger} Email: andreas.veeser{at}mat.unimi.it

§ Corresponding author. Email: marco.verani{at}polimi.it

Received on 9 January 2007. Revised on 2 August 2007.


  Abstract

The dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case, its performance is generally superior than that of global ‘energy norm’ error estimators which are ‘unconditionally’ reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example.

Key Words: a posteriori estimates; output functionals; finite elements; duality


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