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Towards practical anisotropic adaptive FEM on triangular meshes: a new refinement paradigm

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Computing and Visualization in Science

Abstract

Adaptive anisotropic refinement of finite element meshes allows one to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows one to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.

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Notes

  1. This condition may also be formulated in terms of a change in element orientation, if the elements are stored with clockwise ordering of the nodes, for example.

  2. Again, this condition can be formulated in terms of a change in element orientation.

References

  1. Aguilar, J., Goodman, J.: Anisotropic mesh refinement for finite element methods based on error reduction. J. Comput. Appl. Math. 193, 497–515 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  3. Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Teubner, Leipzig (1999)

    Google Scholar 

  4. Apel, T., Grosman, S., Jimack, P., Meyer, A.: A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math. 50, 329–341 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Apel, T., Nicaise, S.: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21(6), 519–549 (1998). doi:10.1002/(SICI)1099-1476(199804)21:6<519::AID-MMA962>3.0.CO;2-R

  6. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)

    Book  MATH  Google Scholar 

  7. Bank, R., Smith, R.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30(4), 921–935 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bänsch, E.: Local mesh refinement in 2 and 3 dimensions. Impact Comput. Sci. Eng. 3, 181–191 (1991)

    Article  MATH  Google Scholar 

  9. Borouchaki, H., George, P., Hecht, F., Laug, P., Saltel, E.: Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Finite Elem. Anal. Des. 25, 61–83 (1997). doi:10.1016/S0168-874X(96)00057-1

    Article  MATH  MathSciNet  Google Scholar 

  10. Bürg, M., Dörfler, W.: Convergence of an adaptive \(hp\) finite element strategy in higher space-dimension. Appl. Numer. Math. 61, 1132–1146 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cao, W.: On the error of linear interpolation and the orientation, aspect ratio, and internal angles of a triangle. SIAM J. Numer. Anal. 43, 19–40 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dolejsi, V.: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1, 165–178 (1998)

    Article  MATH  Google Scholar 

  13. Dörfler, W., Heuveline, V.: Convergence of an adaptive \(hp\) finite element strategy in one space dimension. Appl. Numer. Math. 57, 1108–1124 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  15. Formaggia, L., Micheletti, S., Perotto, S.: 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  MATH  MathSciNet  Google Scholar 

  16. Fortin, M.: Anisotropic mesh adaptation through hierarchical error estimators. In: P. Minev, Y. Lin (eds.) Scientific computing and applications, vol. 7, pp. 53–65. Nova Science Publishers, New York (2001)

  17. Grosman, S.: Adaptivity in anisotropic finite element calculations. Ph.D. thesis, TU-Chemnitz, Chemnitz, Germany (2006)

  18. Habashi, W., Dompierre, J., Bourgault, Y., Ait-Ali-Yahia, D., Fortin, M., Vallet, M.G.: Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independant CFD. Part I: general principles. Int. J. Numer. Meth. Fluids 32, 725–744 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hecht, F.: Bidimensional anisotropic mesh generator. Technical report, INRIA, Rocquencourt (1997). Software: http://www.ann.jussieu.fr/hecht/ftp/bamg/

  20. Huang, W., Kamenski, L., Lang, J.: A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates. J. Comput. Phys. 229(6), 2179–2198 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kelly, D., de Gago, S.R.J., Zienkiewicz, O., Babuska, I.: A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis. Int. J. Numer. Meth. Eng. 19, 1593–1619 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kornhuber, R., Roitzsch, R.: On adaptive grid refinement in the presence of internal or boundary layers. Impact Comput. Sci. Eng. 2, 40–72 (1990)

    Article  Google Scholar 

  23. Kunert, G., Verfürth, R.: Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86, 283–303 (2000)

    Google Scholar 

  24. Li, X., Shephard, M., Beall, M.: 3d anisotropic mesh adaptation by mesh modification. Comput. Methods Appl. Mech. Eng. 194, 4915–4950 (2005)

    Google Scholar 

  25. Mahmood, R., Jimack, P.: Locally optimal unstructured finite element meshes in 3 dimensions. Comput. Struct. 82(23–26), 2105–2116 (2004)

    Article  Google Scholar 

  26. Mitchell, W.: Optimal multilevel iterative methods for adaptive grids. SIAM J. Sci. Comput. 13(1), 146–167 (1992)

    Article  MATH  Google Scholar 

  27. Picasso, M., Alauzet, F., Borouchaki, H., George, P.L.: A numerical study of some Hessian recovery techniques on isotropic and anisotropic meshes. SIAM J. Sci. Comput. 33(3), 1058–1076 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  29. Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, vol. 24, 2nd edn. Springer, Berlin (2008)

  30. Beuchler, S., Meyer, A.: SPC-PM3AdH v1.0 - Programmer’s Manual. Technical report Preprint SFB393/01-08, TU Chemnitz, Chemnitz (2001). Available at http://www.tu-chemnitz.de/sfb393/

  31. Schneider, R.: A review of anisotropic refinement methods for triangular meshes in fem. In: Apel, T., Steinbach, O. (eds.) Advanced Finite Element Methods and Applications, Lecture Notes in Applied and Computational Mechanics, vol. 66, pp. 133–152. Springer, Berlin, (2013). doi:10.1007/978-3-642-30316-6_6

  32. de Gago, S.R., Kelly, D., Zienkiewicz, O., Babuska, I.: A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement. Int. J. Numer. Methods Eng. 19, 1621–1656 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wang, D., Li, R., Yan, N.: An edge-based anisotropic mesh refinement algorithm and its application to interface problems. Commun. Comput. Phys. 8(3), 511–540 (2010). doi:10.4208/cicp.210709.121109a

    MathSciNet  Google Scholar 

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Acknowledgments

We thank all those who have contributed to this work by providing ideas, comments and suggestions during discussions, especially Thomas Apel, Arnd Meyer and Thomas Mach.

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, 09107 , Chemnitz, Germany

    René Schneider

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  1. René Schneider
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Correspondence to René Schneider.

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Communicated by Randolph E. Bank.

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Schneider, R. Towards practical anisotropic adaptive FEM on triangular meshes: a new refinement paradigm. Comput. Visual Sci. 15, 247–270 (2012). https://doi.org/10.1007/s00791-013-0212-5

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  • DOI: https://doi.org/10.1007/s00791-013-0212-5

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