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Computational Methods in Applied Mathematics

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Volume 17, Issue 4


A Partition-of-Unity Dual-Weighted Residual Approach for Multi-Objective Goal Functional Error Estimation Applied to Elliptic Problems

Bernhard Endtmayer / Thomas Wick
  • Corresponding author
  • Centre de Mathématiques Appliquées, École Polytechnique, Université Paris-Saclay, 91128 Palaiseau, France
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Published Online: 2017-04-05 | DOI: https://doi.org/10.1515/cmam-2017-0001


In this work, we design a posteriori error estimation and mesh adaptivity for multiple goal functionals. Our method is based on a dual-weighted residual approach in which localization is achieved in a variational form using a partition-of-unity. The key advantage is that the method is simple to implement and backward integration by parts is not required. For treating multiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete error problem) and suggest an alternative way for its computation. Our algorithmic developments are substantiated for elliptic problems in terms of four different numerical tests that cover various types of challenges, such as singularities, different boundary conditions, and diverse goal functionals. Moreover, several computations with higher-order finite elements are performed.

Keywords: Finite Element Method; Mesh Adaptivity; Dual-Weighted Residual; Partition-of-Unity, Multi-Objective Goal Functionals; Adjoint to the Adjoint Problem

MSC 2010: 65N30; 65M60; 49M15; 35Q74


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About the article

Received: 2016-10-05

Revised: 2017-02-03

Accepted: 2017-02-24

Published Online: 2017-04-05

Published in Print: 2017-10-01

Citation Information: Computational Methods in Applied Mathematics, Volume 17, Issue 4, Pages 575–599, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0001.

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