Advertisement

A Review of Anisotropic Refinement Methods for Triangular Meshes in FEM

  • René Schneider
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 66)

Abstract

This review gives an overview of current anisotropic refinement methods in finite elements, with the focus on the actual refinement step. In this we highlight strengths and weaknesses of different approaches and hope to stimulate research into closer coupling of the refinement process with efficient solution strategies for the equation systems arising from the discretized equations. A rough overview of different categories of error estimation techniques relevant in the anisotropic setting is also given.

Keywords

Posteriori Error Coarse Mesh Triangular Mesh Posteriori Error Estimate Initial Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000)Google Scholar
  2. 2.
    Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Teubner, Leipzig (1999)Google Scholar
  3. 3.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Babuška, I., Aziz, A.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13(2), 214–226 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Baines, M.: Grid adaptation via node movement. Appl. Numer. Math. 26, 77–96 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)zbMATHGoogle Scholar
  7. 7.
    Bank, R., Smith, R.: Mesh smoothing using a posteriori error estimates. SIAM J. Numer. Anal. 34(3), 979–997 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Borouchaki, H., George, P., Hecht, F., Laug, P., Saltel, E.: Delaunay mesh generation governed by metric specifications. part i. algorithms. Finite Elements Anal. Design 25, 61–83 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dolejsi, V.: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1, 165–178 (1998)zbMATHCrossRefGoogle Scholar
  11. 11.
    Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fortin, M.: Anisotropic mesh adaptation through hierarchical error estimators. In: Minev, P., Lin, Y. (eds.) Scientific Computing and Applications, vol. 7, pp. 53–65. Nova Science Publishers (2001)Google Scholar
  14. 14.
    Grajewski, M., Köster, M., Turek, S.: Numerical analysis and implementational aspects of a new multilevel grid deformation method. Appl. Numer. Math. 60(8), 767–781 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Grosman, S.: Adaptivity in anisotropic finite element calculations. PhD thesis, TU Chemnitz, Chemnitz, Germany (2006)Google Scholar
  16. 16.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hecht, F.: Bidimensional anisotropic mesh generator. Tech. rep., INRIA, Rocquencourt, software (1997), http://www.ann.jussieu.fr/hecht/ftp/bamg/
  18. 18.
    Huang, W.: Mathematical principles of anisotropic mesh adaptation. Comm. Comput. Phys. 1(2), 276–310 (2005)Google Scholar
  19. 19.
    Huang, W., Russell, R.: Adaptive Moving Mesh Methods. Applied Mathematical Sciences. Springer (2011)Google Scholar
  20. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kornhuber, R., Roitzsch, R.: On adaptive grid refinement in the presence of internal or boundary layers. Impact Comput. Sci. Engrg. 2, 40–72 (1990)CrossRefGoogle Scholar
  22. 22.
    Kunert, G.: A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes. PhD thesis, TU Chemnitz (1999), http://archiv.tu-chemnitz.de/pub/1999/0012/index.html
  23. 23.
    Kunert, G.: Toward anisotropic mesh construction and error estimation in the finite element method. Numer. Meth. Partial Diff. Eqns. 18(5), 625–648 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Lang, J., Cao, W., Huang, W., Russel, R.: A two-dimensional moving finite element method with local refinement based on a posteriori error estimates. Appl. Numer. Math. 46, 75–94 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Li, X., Shephard, M., Beall, M.: 3d anisotropic mesh adaptation by mesh modification. Comput. Meth. Appl. Mech. Engrg. 194, 4915–4950 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mahmood, R., Jimack, P.: Locally optimal unstructured finite element meshes in 3 dimensions. Comput. Struct. 82(23–26), 2105–2116 (2004)CrossRefGoogle Scholar
  28. 28.
    Miller, K., Miller, R.: Moving finite elements 1. SIAM J. Numer. Anal. 18(6), 1019–1032 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    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)zbMATHCrossRefGoogle Scholar
  31. 31.
    Roos, H.–G.: Layer-adapted grids for singular perturbation problems. ZAMM Z. Angew. Math. Mech. 78(5), 291–309 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Beuchler, S., Meyer, A.: SPC-PM3AdH v1.0 - Programmer’s Manual. Tech. Rep. Preprint SFB393/01-08, TU Chemnitz, Chemnitz (2001), http://www.tu-chemnitz.de/sfb393/
  33. 33.
    Schneider, R.: Applications of the discrete adjoint method in computational fluid dynamics. PhD thesis, University of Leeds (2006), http://www.comp.leeds.ac.uk/research/pubs/theses/schneider.pdf
  34. 34.
    Schneider, R., Jimack, P.: Toward anisotropic mesh adaption based upon sensitivity of a posteriori estimates. School of Computing Research Report Series 2005.03, University of Leeds (2005), http://www.engineering.leeds.ac.uk/computing/research/publications/reports/200407/
  35. 35.
    Shewchuk, J.: What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures. Preprint, Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA (2002), http://www.cs.berkeley.edu/~jrs/papers/elemj.pdf
  36. 36.
    Vallet, M.G., Manole, C.M., Dompierre, J., Dufour, S., Guibault, F.: Numerical comparison of some Hessian recovery techniques. Int. J. Numer. Meth. Engrg. 72(8), 987–1007 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Walkley, M., Jimack, P., Berzins, M.: Anisotropic adaptivity for finite element solutions of 3-d convection-dominated problems. Int. J. Numer. Methods Fluids 40, 551–559 (2002)zbMATHCrossRefGoogle Scholar
  38. 38.
    Wang, D., Li, R., Yan, N.: An edge-based anisotropic mesh refinement algorithm and its application to interface problems. Comm. Comput. Phys. 8(3), 511–540 (2010)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikTU ChemnitzChemnitzGermany

Personalised recommendations