Goal-oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Vincent Darrigrand*, Ángel Rodríguez-Rozas, Ignacio Muga, David Pardo, Albert Romkes, Serge Prudhomme

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


In goal-oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal-oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two- and three-dimensional (2-D and 3-D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.

Original languageEnglish (US)
Pages (from-to)22-42
Number of pages21
JournalInternational Journal for Numerical Methods in Engineering
Issue number1
StatePublished - Jan 6 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
Copyright © 2017 John Wiley & Sons, Ltd.


  • Helmholtz equation
  • error representation
  • finite element methods
  • goal-oriented adaptivity

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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