Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


IMPACT FACTOR 2018: 1.218
5-year IMPACT FACTOR: 1.411

CiteScore 2018: 1.42

SCImago Journal Rank (SJR) 2018: 0.947
Source Normalized Impact per Paper (SNIP) 2018: 0.939

Mathematical Citation Quotient (MCQ) 2017: 0.76

Online
ISSN
1609-9389
See all formats and pricing
More options …
Volume 17, Issue 4

Issues

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-04-05 | DOI: https://doi.org/10.1515/cmam-2017-0001

Abstract

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

References

  • [1]

    M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg. 142 (1997), no. 1–2, 1–88. CrossrefGoogle Scholar

  • [2]

    M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (New York), John Wiley & Sons, New York, 2000. Google Scholar

  • [3]

    G. Allaire, Analyse numerique et optimisation, Les Éditions de l’École Polytechnique, Palaiseau, 2005. Google Scholar

  • [4]

    J. Andersson and H. Mikayelyan, The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford–Shah functional in the plain. A revision, preprint (2015), https://arxiv.org/abs/1204.5328v2.

  • [5]

    T. Apel, A.-M. Saendig and J. R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci. 19 (1996), no. 1, 63–85. CrossrefGoogle Scholar

  • [6]

    W. Bangerth, R. Hartmann and G. Kanschat, deal.II – A general purpose object oriented finite element library, ACM Trans. Math. Softw. 33 (2007), no. 4, Article ID 24. Web of ScienceGoogle Scholar

  • [7]

    W. Bangerth, T. Heister and G. Kanschat, deal. II Differential equations analysis library, Technical Reference (2013). Google Scholar

  • [8]

    W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Math. ETH Zürich, Birkhäuser, Basel, 2003. Google Scholar

  • [9]

    R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, ENUMATH’97 (Heidelberg 1997), World Scientific, Singapore (1998), 621–637. Google Scholar

  • [10]

    R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001), 1–102. Google Scholar

  • [11]

    A. Bonnet and G. David, Cracktip is a Global Mumford–Shah Minimizer, Astérisque 274, Société Mathématique de France, Paris, 2001. Google Scholar

  • [12]

    M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul. 1 (2003), no. 2, 221–238. CrossrefGoogle Scholar

  • [13]

    D. Braess, Finite Elemente. Theory, Fast Solvers and Applications in Elasticity Theory, Springer, Berlin, 2007. Google Scholar

  • [14]

    G. F. Carey and J. T. Oden, Finite Elements. Volume III: Computational Aspects, Texas Finite Elem., Prentice-Hall, Englewood Cliffs, 1984. Google Scholar

  • [15]

    C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 5, 1571–1587. CrossrefGoogle Scholar

  • [16]

    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1987. Google Scholar

  • [17]

    M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen, Springer, Berlin, 2006. Google Scholar

  • [18]

    K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica 1995, Cambridge University Press, Cambridge (1995), 105–158. Google Scholar

  • [19]

    D. Estep, M. Holst and M. Larson, Generalized green’s functions and the effective domain of influence, SIAM J. Sci. Comput. 26 (2005), no. 4, 1314–1339. CrossrefGoogle Scholar

  • [20]

    L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 2010. Google Scholar

  • [21]

    M. Giles and E. Süli, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numer. 11 (2002), 145–236. Google Scholar

  • [22]

    C. Großmann, H.-G. Roos and M. Stynes, Numerical Treatment of Partial Differential Equations, Springer, Berlin, 2007. Google Scholar

  • [23]

    R. Hartmann, Multitarget error estimation and adaptivity in aerodynamic flow simulations, SIAM J. Sci. Comput. 31 (2008), no. 1, 708–731. CrossrefWeb of ScienceGoogle Scholar

  • [24]

    R. Hartmann and P. Houston, Goal-oriented a posteriori error estimation for multiple target functionals, Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin (2003), 579–588. Google Scholar

  • [25]

    P. Houston, B. Senior and E. Sueli, hp-discontinuous galerkin finite element methods for hyperbolic problems: Error analysis and adaptivity, Internat. J. Numer. Methods Fluids 40 (2002), no. 1–2, 153–169. CrossrefGoogle Scholar

  • [26]

    D. Kuzmin and S. Korotov, Goal-oriented a posteriori error estimates for transport problems, Math. Comput. Simulation 80 (2010), no. 8, 1674–1683. Web of ScienceCrossrefGoogle Scholar

  • [27]

    J. Oden and S. Prudhomme, On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg. 176 (1999), 313–331. CrossrefGoogle Scholar

  • [28]

    J. Peraire and A. Patera, Bounds for linear-functional outputs of coercive partial differential Equations: Local indicators and adaptive refinement, Advances in Adaptive Computational Methods in Mechanics, Elsevier, Amsterdam (1998), 199–215. Google Scholar

  • [29]

    R. Rannacher and F.-T. Suttmeier, A feed-back approach to error control in finite element methods: Application to linear elasticity, Comput. Mech. 19 (1997), no. 5, 434–446. CrossrefGoogle Scholar

  • [30]

    T. Richter, Goal-oriented error estimation for fluid-structure interaction problems, Comput. Methods Appl. Mech. Engrg. 223–224 (2012), 38–42. Google Scholar

  • [31]

    T. Richter and T. Wick, Variational localizations of the dual weighted residual estimator, J. Comput. Appl. Math. 279 (2015), no. 0, 192–208. CrossrefWeb of ScienceGoogle Scholar

  • [32]

    A. Schroeder and A. Rademacher, Goal-oriented error control in adaptive mixed FEM for signorini’s problem, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 1–4, 345–355. CrossrefGoogle Scholar

  • [33]

    E. van Brummelen, S. Zhuk and G. van Zwieten, Worst-case multi-objective error estimation and adaptivity, Comput. Methods Appl. Mech. Engrg. 313 (2017), 723–743. CrossrefGoogle Scholar

  • [34]

    R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley, New York, 1996. Google Scholar

  • [35]

    S. Weisser and T. Wick, The dual-weighted residual estimator realized on polygonal meshes, Preprint Number 384, Saarland University, Department of Mathematics, 2016, https://www.math.uni-sb.de/service/preprints/.

  • [36]

    T. Wick, Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity, Comput. Mech. 57 (2016), no. 6, 1017–1035. Web of ScienceCrossrefGoogle Scholar

  • [37]

    J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. Google Scholar

  • [38]

    K. Zee, E. Brummelen, I. Akkerman and R. Borst, Goal-oriented error estimation and adaptivity for fluid-structure interaction using exact linearized adjoints, Comput. Methods Appl. Mech. Engrg. 200 (2011), 2738–2757. CrossrefGoogle Scholar

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.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Kenan Kergrene, Serge Prudhomme, Ludovic Chamoin, and Marc Laforest
Computer Methods in Applied Mechanics and Engineering, 2017

Comments (0)

Please log in or register to comment.
Log in