Skip to main content
SpringerLink
Log in

Error Estimation and Adaptivity

  • Chapter
  • First Online:
Fluid-structure Interactions

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 118))

  • 2503 Accesses

Abstract

This chapter is devoted to a posteriori error estimation and adaptivity. Error estimators can help to control the quality of simulation results and serve as stopping criteria for our algorithms. In the following section we will start by gathering the basics of a posteriori error estimation in the finite element method. As we aim at the application to complex problems like fluid-structure interactions, the main target will be efficiency and ease of realization. Error estimation will be based on the Dual Weighted Residual Method, that has been developed by Becker and Rannacher [40, 41] as a very flexible tool to estimate errors in goal functionals of the solution that can be any kind of output values, such as the deformation of a solid-point, the wall stress on an elastic obstacle or the vorticity in a flow field.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Ainsworth, J.T. Oden, A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65(1), 23–50 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Ainsworth, J.T. Oden, A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142(1–2), 1–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Alauzet, W. Hassan, M. Picasso, Goal oriented, anisotropic, a posteriori error estimates for the laplace equation, in Proceedings of ENUMATH 2009, the 8th European Conference on Numerical Mathematics and Advanced Applications, Uppsala, July 2009, ed. by G. Kreiss, P. Lötstedt, M. Neytcheva, A. Målqvist (Springer, Berlin, 2010), pp. 47–58

    Google Scholar 

  4. T. Apel, Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics (Teubner, Stuttgart, 1999)

    Google Scholar 

  5. I. Babuška, Feedback, adaptivity and a posteriori estimates in finite elements, aims, theory, and experience, in Accuracy Estimates and Adaptive Refinements in Finite Element Computations, ed. by I. Babuška et al. (Wiley, New York, 1986), pp. 3–23

    Google Scholar 

  6. I. Babuška, A.D. Miller, The post-processing approach in the finite element method. I. calculations of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Methods Eng. 20, 1085–1109 (1984)

    Google Scholar 

  7. I. Babuška, A.D. Miller, The post-processing approach in the finite element method. III. a posteriori error estimation and adaptive mesh selection. Int. J. Numer. Methods Eng. 20, 2311–2324 (1984)

    MATH  Google Scholar 

  8. I. Babuška, W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. R.E. Bank, A.H. Sherman, A. Weiser, Some refinement algorithms and data structures for regular local mesh refinement, in Scientific Computing. Applications of Mathematics and Computing to the Physical Sciences, ed. by Stepleman et al. Transactions on Scientific Computing, vol. 1 (North-Holland, Amsterdam, 1983), pp. 3–17

    Google Scholar 

  10. R.E. Barnhill, J.A. Gregory Interpolation remainder theory from Taylor expansions on triangle. Numer. Math. 25, 401–408 (1976)

    Google Scholar 

  11. P. Bastian, Parallel adaptive multigrid methods. Technical Report 93–60, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, 1993

    Google Scholar 

  12. R. Becker, Weighted error estimators for finite element approximations of the incompressible Navier-Stokes equations. Rapport de Recherche RR-3458, INRIA Sophia-Antipolis, 1998

    Google Scholar 

  13. R. Becker, M. Braack, Solution of a benchmark problem for natural convection at low mach number natural convection at low mach number. SFB Preprint, 30, 2000

    Google Scholar 

  14. R. Becker, R. Rannacher, Weighted a posteriori error control in FE methods, in ENUMATH’97, ed. by H.G. Bock et al. (World Science Publisher, Singapore, 1995)

    Google Scholar 

  15. R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, in Acta Numerica 2001, vol. 37, ed. by A. Iserles (Cambridge University Press, Cambridge, 2001), pp. 1–225

    Google Scholar 

  16. R. Becker, B. Vexler, A posteriori error estimation for finite element discretization of parameter identification problems. Numer. Math. 96(3), 435–459 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Becker, M. Braack, R. Rannacher, On error control for reactive flow problems, in Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties. Scientific Computing in Chemical Engineering II, vol. 1 (Springer, Berlin, 1999), pp. 320–327

    Google Scholar 

  18. R. Becker, D. Meidner, B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22(5), 813–833 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Bengzon, M.G. Larson, Adaptive finite element approximation of multiphysics problems: a fluid-structure interaction model problem. Int. J. Numer. Methods Eng. 84, 1451–1465 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Besier, R. Rannacher, Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. Int. J. Numer. Math. Fluids. 70(9), 1139–1166 (2012)

    Article  MathSciNet  Google Scholar 

  21. H. Blum, Asymptotic error expansion and defect correction in the finite element method. Habilitationsschrift, Institut für Angewandte Mathematik, Universität Heidelberg, 1991. SFB-123 Preprint 640

    Google Scholar 

  22. H. Blum, Q. Lin, R. Rannacher, Asymptotic error expansion and Richardson extrapolation for linear finite elements. Numer. Math. 49, 11–37 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Braack, An adaptive finite element method for reactive flow problems, Ph.D. thesis, Universität Heidelberg, 1998

    Google Scholar 

  24. M. Braack, A. Ern, A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1(2), 221–238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Braack, A. Ern, Adaptive computation of reactive flows with local mesh refinement and model adaptation, in Numerical Mathematics and Advanced Applications, ENUMATH 2003, ed. by M. Feistauer et al. (Springer, Berlin, 2004), pp. 159–168

    Google Scholar 

  26. M. Braack, T. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements. Comput. Fluids 35(4), 372–392 (2006)

    Article  MATH  Google Scholar 

  27. M. Braack, T. Richter, Stabilized finite elements for 3-d reactive flows. Int. J. Numer. Math. Fluids 51, 981–999 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Braack, N. Taschenberger, A posteriori control of modeling and discretization errors for quasi periodic solutions. J. Numer. Math. 22(2), 87–108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Braack, R. Becker, R. Rannacher, Adaptive finite elements for reactive flows, in ENUMATH-97, Second European Conference on Numerical Mathematics and Advanced Applications, Enumath (World Scientific Publisher, Singapore, 1998), pp. 206–213

    MATH  Google Scholar 

  30. M. Braack, E. Burman, N. Taschenberger, Duality based a posteriori error estimation for quasi-periodic solutions using time averages. SIAM J. Sci. Comput. 33, 2199–2216 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. H.-J. Bungartz, M. Schäfer (eds.), Fluid-Structure Interaction. Modelling, Simulation, Optimisation. Lecture Notes in Computational Science and Engineering, vol. 53 (Springer, Berlin, 2006). ISBN-10: 3-540-34595-7

    Google Scholar 

  32. H.-J. Bungartz, M. Schäfer (eds.), Fluid-Structure Interaction II. Modelling, Simulation, Optimisation. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2010)

    Google Scholar 

  33. G.F. Carey, S.S. Chow, M.K. Seager, Approximate boundary-flux calculations. Comput. Methods Appl. Mech. Eng. 50, 107–120 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Carpio, J.L. Prieto, R. Bermejo, Anisotropic “goal-oriented” mesh adaptivity for elliptic problems. SIAM J. Sci. Comput. 35(2), A861–A885 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. C. Carstensen, Convergence of adaptive finite element methods in computational mechanics. Appl. Numer. Math. 59, 2119–2130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. M.J. Castro-Diaz, F. Hecht, B. Mohammadi, O. Pironneau, Anisotropic unstructured mesh adaptation for flow simulations. Int. J. Numer. Math. Fluids 25, 475–491 (1997)

    Article  MATH  Google Scholar 

  37. E.F. D’Azevedo, R.B. Simpson, On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10(6), 1063–1075 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations. SIAM J. Numer. Anal. 32(1), 1–48 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. L. Failer, Optimal control for time dependent nonlinear fluid-structure interaction, Ph.D. thesis, Technische Universität München, 2017

    Google Scholar 

  41. L. Failer, D. Meidner, B. Vexler, Optimal control of a linear unsteady fluid-structure interaction problem. J. Optim. Theory Appl. 170(1), 1–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Formaggia, S. Perotto, P. Zunino, An anisotropic a-posteriori error estimate for a convection-diffusion problem. Comput. Visual. Sci. 4, 99–2001 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. L. Formaggia, S. Micheletti, S. Perotto, Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the stokes problems. Appl. Numer. Math. 51(4), 511–533 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. M.B. Giles, E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, in Acta Numerica 2002, ed. by A. Iserles (Cambridge University Press, Cambridge, 2002), pp. 145–236

    Chapter  Google Scholar 

  45. W. Hackbusch, Multi-Grid Methods and Applications (Springer, Berlin, 1985)

    Book  MATH  Google Scholar 

  46. R. Hartmann, Adaptive FE-methods for conservation equations, in Eighth International Conference on Hyperbolic Problems. Theory, Numerics, Applications (HYP2000), ed. by G. Warnecke (Birkhauser, Basel, 2000)

    Google Scholar 

  47. R. Hartmann, Multitarget error estimation and adaptivity in aerodynamic flow simulations. SIAM J. Sci. Comput. 31(1), 708–731 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Hassan, Algorithmes d’adaptation de maillages anisotropes et application á l’aérodynamique, Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2012. doi:10.5075/epfl-thesis-5304

    Google Scholar 

  49. V. Heuveline, R. Rannacher, Duality-based adaptivity in the hp-finite element method. J. Numer. Math. 11(2), 95–113 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Holst, S. Pollock, Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems. Numer. Methods Partial Differ. Equ. 32(2), 479–509 (2015, online)

    Google Scholar 

  51. J. Hron, S. Turek, M. Madlik, M. Razzaq, H. Wobker, J.F. Acker, Numerical simulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics, in Fluid-Structure Interaction II: Modeling, Simulation, Optimization, ed. by H.-J. Bungartz, M. Schäfer. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2010), pp. 197–220

    Google Scholar 

  52. C. Johnson, A. Szepessy, Adaptive finite element methods for conservation laws based on a posteriori error estimates. Commun. Pure Appl. Math. 48(3), 199–234 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. T. Leicht, R. Hartmann, Anisotropic mesh refinement for discontinuous Galerkin methods in two-dimensional aerodynamic flow simulations. Int. J. Numer. Math. Fluids 56, 2111–2138 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. L. Machiels, A.T. Patera, J. Peraire, Output bound approximation for partial differential equations; application to the incompressible Navier-Stokes equations, in Industrial and Environmental Applications of Direct and Large Eddy Numerical Simulation, ed. by S. Biringen (Springer, Berlin/Heidelberg/New York, 1998)

    Google Scholar 

  55. D. Meidner, Adaptive space-time finite element methods for optimization problems governed by nonlinear parabolic systems, Ph.D. thesis, University of Heidelberg, 2008

    Google Scholar 

  56. D. Meidner, T. Richter, Goal-oriented error estimation for the fractional step theta scheme. Comput. Methods Appl. Math. 14, 203–230 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  57. D. Meidner, T. Richter, A posteriori error estimation for the fractional step theta discretization of the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 288, 45–59 (2015)

    Article  MathSciNet  Google Scholar 

  58. P. Morin, R.H. Nochetto, K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  59. R.H. Nochetto, A. Veeser, M. Verani, A safeguarded dual weighted residual method. IMA J. Numer. Anal. 29(1), 126–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. C. Nystedt, A priori and a posteriori error estimates and adaptive finite element methods for a model eigenvalue problem. Technical Report Preprint NO 1995-05, Department of Mathematics, Chalmers University of Technology, 1995

    Google Scholar 

  61. J.T. Oden, K. Vemaganti, Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; part II: a computational environment for adaptive modeling of heterogeneous elastic solids. Comput. Methods Appl. Mech. Eng. 190, 6089–6124 (2001)

    Article  MATH  Google Scholar 

  62. J. Peraire, M. Vahdati, K. Morgan, O.C. Zienkiewicz, Adaptive remeshing for compressible flow computations. J. Comput. Phys. 72, 449–466 (1987)

    Article  MATH  Google Scholar 

  63. R. Rannacher, F.-T. Suttmeier, A posteriori error control in finite element methods via duality techniques: Application to perfect plasticity. Comput. Mech. 21, 123–133 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  64. R. Rannacher, F.-T. Suttmeier, A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Eng. 176, 333–361 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  65. T. Richter, Funktionalorientierte Gitteroptimierung bei der Finite-Elemente Approximation elliptischer Differentialgleichungen. Diplomarbeit, Universität Heidelberg, June 2001

    Google Scholar 

  66. T. Richter, Parallel multigrid for adaptive finite elements and its application to 3D flow problem, Ph.D. thesis, Universität Heidelberg, 2005. URN:nbn:de:bsz:16-opus-57433

    Google Scholar 

  67. T. Richter, A posteriori error estimation and anisotropy detection with the dual weighted residual method. Int. J. Numer. Math. Fluids 62(1), 90–118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  68. T. Richter, Goal oriented error estimation for fluid-structure interaction problems. Comput. Methods Appl. Mech. Eng. 223/224, 28–42 (2012)

    Google Scholar 

  69. T. Richter, Anisotropic finite elements for fluid-structure interactions, in Numerical Mathematics and Advanced Applications-ENUMATH 2011 (Springer, Berlin, 2013), pp. 73–70

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  71. T. Richter, A. Springer, B. Vexler, Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems. Numer. Math. 124(1), 151–182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  72. M. Schäfer, S. Turek, Benchmark computations of laminar flow around a cylinder. (With support by F. Durst, E. Krause and R. Rannacher), in Flow Simulation with High-Performance Computers II. DFG Priority Research Program Results 1993–1995, ed. by E.H. Hirschel. Notes on Numerical Fluid Mechanics, vol. 52 (Vieweg, Wiesbaden, 1996), pp. 547–566

    Google Scholar 

  73. M. Schmich, Adaptive finite element methods for computing nonstationary incompressible flows, Ph.D. thesis, Universität Heidelberg, 2009. URN:nbn:de:bsz:16-opus-102001

    Google Scholar 

  74. M. Schmich, B. Vexler, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369–393 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Seitz, Vergleich von Teilschritt-Theta-Verfahren, Master’s thesis, Universität Heidelberg, 2013. Diploma’s thesis, Universität Heidelberg

    Google Scholar 

  76. A. Springer, Efficient higher order discontinuous Galerkin time discretizations for parabolic optimal control problems, Ph.D. thesis, Technische Universität München, 2015. URN:nbn:de:bvb:91-diss-20150519-1237294-1-6

    Google Scholar 

  77. R. Stevenson, An optimal adaptive finite element method. SIAM J. Numer. Anal. 42(5), 2188–2217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  78. S. Turek, J. Hron, M. Madlik, M. Razzaq, H. Wobker, J. Acker, Numerical simulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics. Technical report, Fakultät für Mathematik, TU Dortmund, Feb 2010. Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 403

    Google Scholar 

  79. D.A. Venditti, D.L. Darmofal, Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows. J. Comput. Phys. 187, 22–46 (2003)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  81. B. Vexler, W. Wollner, Adaptive finite elements for elliptic optimization problems with control constrains. SIAM J. Contin. Opt. 47, 509–534 (2008)

    Article  MATH  Google Scholar 

  82. O.C. Zienkiewicz, J. Wu, Automatic directional refinement in adaptive analysis of compressible flows. Int. J. Numer. Methods Eng. 37, 2189–2210 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Für Analysis und Numerik, Universität Magdeburg, Magdeburg, Germany

    Thomas Richter

Authors
  1. Thomas Richter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Richter, T. (2017). Error Estimation and Adaptivity. In: Fluid-structure Interactions. Lecture Notes in Computational Science and Engineering, vol 118. Springer, Cham. https://doi.org/10.1007/978-3-319-63970-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation