Multigoal-oriented error estimates for non-linear problems

Bernhard Endtmayer 1 , Ulrich Langer 1  and Thomas Wick 2
  • 1 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040, Linz, Austria
  • 2 Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany
Bernhard Endtmayer
  • Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040, Linz, Austria
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, Ulrich Langer
  • Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040, Linz, Austria
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and Thomas Wick
  • Corresponding author
  • Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany
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Abstract

In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

  • [1]

    R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Elsevier Science, 2003.

  • [2]

    M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, 2000.

  • [3]

    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, arXiv: 1205.5328v2, 2015.

  • [4]

    D. N. Arnold, D. Boffi, and R. S. Falk, Approximation by quadrilateral finite elements, Math. Comp. 71 (2002), 909 922.

    • Crossref
    • Export Citation
  • [5]

    W. Bangerth, D. Davydov, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, and D. Wells, The deal.II library, version 8.4, J. Numer. Math. 24 (2016), 135 141.

  • [6]

    W. Bangerth, R. Hartmann, and G. Kanschat, deal.II   a general purpose object oriented finite element library, ACM Trans. Math. Softw. 33 (2007), 24/1 27.

  • [7]

    W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhäuser Verlag, Boston, 2003.

  • [8]

    R. Becker, C. Johnson, and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing 55 (1995), 271 288.

    • Crossref
    • Export Citation
  • [9]

    R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, In: ENUMATH'97 (Eds. H. G. Bock et al.), World Sci. Publ., Singapore, 1995.

  • [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.

    • Crossref
    • Export Citation
  • [11]

    A. Bonnet and G. David, Cracktip is a Global Mumford-Shah Minimizer, Asterisque No. 274, 2001.

  • [12]

    M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul. 1 (2003), 221 238.

    • Crossref
    • Export Citation
  • [13]

    D. Braess, Finite Elemente; Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, 4., überarbeitete und erweiterte Auflage ed, Springer-Verlag, Berlin Heidelberg, 2007.

  • [14]

    D. Breit, A. Cianchi, L. Diening, T. Kuusi, and S. Schwarzacher, The p-Laplace system with right-hand side in divergence form: inner and up to the boundary pointwise estimates, Nonlinear Anal. 153 (2017), 200 212.

    • Crossref
    • Export Citation
  • [15]

    G. F. Carey and J. T. Oden, Finite Elements. Volume III. Computational Aspects, The Texas Finite Element Series, Prentice-Hall, Inc., Englewood Cliffs, 1984.

  • [16]

    C. Carstensen and R. Klose, A posteriori finite element error control for the p-Laplace problem, SIAM J. Sci. Comput. 25 (2003), 792 814.

    • Crossref
    • Export Citation
  • [17]

    P. G. Ciarlet, Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002.

  • [18]

    T. A. Davis, Algorithm 832: UMFPACK V4.3   an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw. 30 (2004), 196 199.

    • Crossref
    • Export Citation
  • [19]

    P. Deuflhard, Newton Methods for Nonlinear Problems, Springer Series in Computational Mathematics, Vol. 35, Springer, Berlin Heidelberg, 2011.

  • [20]

    L. Diening and M. Růžička, Interpolation operators in Orlicz-Sobolev spaces, Numer. Math. 107 (2007), 107 129.

    • Crossref
    • Export Citation
  • [21]

    B. Endtmayer and T. Wick, A partition-of-unity dual-weighted residual approach for multi-objective goal functional error estimation applied to elliptic problems, Comput. Methods Appl. Math. 17 (2017), 575 599.

    • Crossref
    • Export Citation
  • [22]

    A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput. 35 (2013), A1761 A1791.

    • Crossref
    • Export Citation
  • [23]

    C. Giannelli, B. Jüttler, and H. Speleers, THB-splines: the truncated basis for hierarchical splines, Comput. Aided Geom. Design 29 (2012), 485 498.

    • Crossref
    • Export Citation
  • [24]

    R. Glowinski and A. Marroco, Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires, ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 9 (1975), 41 76.

  • [25]

    W. Hackbusch, Multigrid Methods and Applications, Springer, 2003.

  • [26]

    W. Han, A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations, Springer, 2005.

  • [27]

    R. Hartmann, Multitarget error estimation and adaptivity in aerodynamic flow simulations, SIAM J. Sci. Comput. 31 (2008), 708 731.

    • Crossref
    • Export Citation
  • [28]

    R. Hartmann and P. Houston, Goal-oriented a posteriori error estimation for multiple target functionals, In: Proc. of Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin, 2003, pp. 579 588.

  • [29]

    A. Hirn, Finite element approximation of singular power-law systems, Math. Comp. 82 (2013), 1247 1268.

    • Crossref
    • Export Citation
  • [30]

    P. Houston, B. Senior, and E. Süli, $hp$-discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity, Int. J. Numer. Methods Fluids 40 (2002), 153 169,

    • Crossref
    • Export Citation
  • [31]

    T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), 4135 4195.

    • Crossref
    • Export Citation
  • [32]

    K. Kergrene, S. Prudhomme, L. Chamoin, and M. Laforest, A new goal-oriented formulation of the finite element method, Comput. Methods Appl. Mech. Engrg. 327 (2017), 256 276.

    • Crossref
    • Export Citation
  • [33]

    M. A. Khamsi and W. A. Kirk, An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics, Wiley-Interscience, New York, 2001.

  • [34]

    S. K. Kleiss and S. K. Tomar, Guaranteed and sharp a posteriori error estimates in isogeometric analysis, Comput. Math. Appl. 70 (2015), 167 190.

    • Crossref
    • Export Citation
  • [35]

    U. Langer, S. Matculevich, and S. Repin, Functional type error control for stabilised space-time IgA approximations to parabolic problems, In: Large-Scale Scientific Computing (LSSC 2017) (Eds. I. Lirkov and S. Margenov), Lecture Notes in Computer Science (LNCS), Springer-Verlag, 2017, pp. 57 66.

  • [36]

    U. Langer, S. Matculevich and S. Repin, Guaranteed error control bounds for the stabilised space-time IgA approximations to parabolic problem, arXiv:1712.06017 [math.NA], 2017.

  • [37]

    P. Lindqvist, Notes on the p-Laplace equation, Report No. 102, University of Jyväskylä, Department of Mathematics and Statistics, 2006.

  • [38]

    W. B. Liu and J. W. Barrett, A further remark on the regularity of the solutions of the p-Laplacian and its applications to their finite element approximation, Nonlinear Analysis: Theory, Methods and Applications 21 (1993), 379 387.

    • Crossref
    • Export Citation
  • [39]

    W. B. Liu and J. W. Barrett, Higher-order regularity for the solutions of some degenerate quasilinear elliptic equations in the plane, SIAM J. Math. Anal. 24 (1993), 1522 1536.

    • Crossref
    • Export Citation
  • [40]

    W. B. Liu and J. W. Barrett, A remark on the regularity of the solutions of the p-Laplacian and its application to their finite element approximation, J. Math. Anal. Appl. 178 (1993), 470 487.

    • Crossref
    • Export Citation
  • [41]

    D. Meidner, R. Rannacher, and J. Vihharev, Goal-oriented error control of the iterative solution of finite element equations, J. Numer. Math. 17 (2009), 143 172.

  • [42]

    P. Neittaanmäki and S. Repin, Reliable Methods for Computer Simulation: Error Control and Posteriori Estimates, Elsevier, Amsterdam, 2004.

  • [43]

    J. Nocedal and S. J. Wright, Numerical Optimization, Springer Ser. Oper. Res. Financial Engrg., 2006.

  • [44]

    D. Pardo, Multigoal-oriented adaptivity for hp-finite element methods, Procedia Computer Science 1 (2010), 1953 1961.

    • Crossref
    • Export Citation
  • [45]

    R. Rannacher and J. Vihharev, Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors, J. Numer. Math. 21 (2013), 23 61.

  • [46]

    R. Rannacher, A. Westenberger, and W. Wollner, Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error, J. Numer. Math. 18 (2010), 303 327.

  • [47]

    S. Repin, A Posteriori Estimates for Partial Differential Equations, Radon Series on Computational and Applied Mathematics, Vol. 4, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.

  • [48]

    A. Reusken, Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems, Numer. Math. 52 (1988), 251 277.

  • [49]

    T. Richter and T. Wick, Variational localizations of the dual weighted residual estimator, J. Comput. Appl. Math. 279 (2015), 192 208.

    • Crossref
    • Export Citation
  • [50]

    W. Sierpinski, General Topology, Mathematical Expositions, No. 7, University of Toronto Press, Toronto, 1952.

  • [51]

    P. Stolfo, A. Rademacher, and A. Schröder, Dual weighted residual error estimation for the finite cell method, Fakultät für Mathematik, TU Dortmund, Report, September 2017, Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 576.

  • [52]

    I. Toulopoulos and T. Wick, Numerical methods for power-law diffusion problems, SIAM J. Sci. Comput. 39 (2017), A681 A710.

    • Crossref
    • Export Citation
  • [53]

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

    • Crossref
    • Export Citation
  • [54]

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

  • [55]

    S. Weisser and T. Wick, The dual-weighted residual estimator realized on polygonal meshes, Comput. Methods Appl. Math., 18 (2017), No. 4, 753 776.

  • [56]

    T. Wick, Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity, Comput. Mech. 57 (2016), 1017 1035.

    • Crossref
    • Export Citation
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