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Adaptive Finite Elements with Anisotropic Meshes

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Numerical Mathematics and Advanced Applications 2011

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Abstract

The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several difficulties of a corner singularity, a peak, a boundary layer, and a wavefront. Numerical experiment clearly shows the advantage of anisotropic mesh adaptation. The conditioning of the resulting linear equation system is addressed as well. In particular, it is shown that the conditioning with adaptive anisotropic meshes is not as bad as generally assumed.

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Notes

  1. 1.

    In this paper the aspect ratio of a triangular element is defined as the longest edge divided by the shortest altitude. For example, an equilateral triangle has an aspect ratio of \(2/\sqrt{3} \approx 1.15\).

  2. 2.

    Note that \(O({N}^{-0.5}) = O(h)\) for quasi-uniform meshes in 2D.

References

  1. A. Agouzal, K. Lipnikov, and Y. Vassilevski. Generation of quasi-optimal meshes based on a posteriori error estimates. In Proceedings of the 16th International Meshing Roundtable, pages 139–148, 2008.

    Google Scholar 

  2. A. Agouzal, K. Lipnikov, and Y. Vassilevski. Anisotropic mesh adaptation for solution of finite element problems using hierarchical edge-based error estimates. In Proceedings of the 18th International Meshing Roundtable, pages 595–610, 2009.

    Google Scholar 

  3. A. Agouzal, K. Lipnikov, and Y. Vassilevski. Hessian-free metric-based mesh adaptation via geometry of interpolation error. Comput. Math. Math. Phys., 50(1):124–138, Jan. 2010.

    Article  MathSciNet  Google Scholar 

  4. T. Apel, S. Grosman, P. K. Jimack, and A. Meyer. A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math., 50(3–4):329–341, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Cao, W. Huang, and R. D. Russell. Comparison of two-dimensional r-adaptive finite element methods using various error indicators. Math. Comput. Simulation, 56(2):127–143, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. F. D’Azevedo. Optimal triangular mesh generation by coordinate transformation. SIAM J. Sci. Stat. Comput., 12(4):755–786, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. F. D’Azevedo, C. H. Romine, and J. M. Donato. Coefficient adaptive trianglation for strongly anisotroic problems. Technical Report ORNL/TM-13086, Oak Ridge National Laboratory, 1997.

    Google Scholar 

  8. M. Dobrowolski, S. Gräf, and C. Pflaum. On a posteriori error estimators in the finite element method on anisotropic meshes. Electron. Trans. Numer. Anal., 8:36–45, 1999.

    MathSciNet  MATH  Google Scholar 

  9. V. Dolejší. Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci., 1(3):165–178, 1998.

    Article  MATH  Google Scholar 

  10. L. Formaggia, S. Micheletti, and 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. Applied Scientific Computing: Advances in Grid Generatuion, Approximation and Numerical Modeling.

    Google Scholar 

  11. L. Formaggia and S. Perotto. New anisotropic a priori error estimates. Numer. Math., 89(4):641–667, 2001.

    MathSciNet  MATH  Google Scholar 

  12. L. Formaggia and S. Perotto. Anisotropic error estimates for elliptic problems. Numer. Math., 94(1):67–92, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Fröhlich, J. Lang, and R. Roitzsch. Selfadaptive finite element computations with smooth time controller and anisotropic refinement. In Numerical Methods in Engineering ’96, pages 523–527. John Wiley & Sons, New York, 1996.

    Google Scholar 

  14. E. H. Georgoulis, E. Hall, and P. Houston. Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM J. Sci. Comput., 30(1):246–271, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Grosman. Adaptivity in anisotriopic finite element calculations. PhD thesis, Technische Universität München, 2006.

    Google Scholar 

  16. F. Hecht. BAMG. http://www.ann.jussieu.fr/hecht/ftp/bamg/.

  17. W. Huang. Metric tensors for anisotropic mesh generation. J. Comput. Phys., 204(2):633–665, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Huang. Mathematical principles of anisotropic mesh adaptation. Commun. Comput. Phys., 1(2):276–310, 2006.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Huang and X. Li. An anisotropic mesh adaptation method for the finite element solution of variational problems. Finite Elem. Anal. Des., 46(1–2):61–73, 2010.

    Article  MathSciNet  Google Scholar 

  21. L. Kamenski. Anisotropic Mesh Adaptation Based on Hessian Recovery and A Posteriori Error Estimates. PhD thesis, TU Darmstadt, 2009.

    Google Scholar 

  22. L. Kamenski, W. Huang, and H. Xu. Conditioning of finite element equations with arbitrary anisotropic meshes. Submitted, e-print: arXiv:1201.3651, 2012.

    Google Scholar 

  23. X. Li and W. Huang. An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems. J. Comput. Phys., 229(21):8072–8094, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. F. Mitchell. A collection of 2d elliptic problems for testing adaptive algorithms. Technical Report NISTIR 7668, National Institute of Standards and Technology, 2010.

    Google Scholar 

  25. M. Picasso. An anisotropic error indicator based on Zienkiewicz–Zhu error estimator: Application to elliptic and parabolic problems. SIAM J. Sci. Comput., 24(4):1328–1355, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Picasso. Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives. Comput. Methods Appl. Mech. Engrg., 196(1–3):14–23, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. B. Simpson. Anisotropic mesh transformations and optimal error control. Appl. Numer. Math., 14(1–3):183 – 198, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Vassilevski and K. Lipnikov. An adaptive algorithm for quasioptimal mesh generation. Comput. Math. Math. Phys., 39(9):1468–1486, 1999.

    MathSciNet  Google Scholar 

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Acknowledgements

This research was supported in part by the National Science Foundation (U.S.A.) through grant DMS-1115118 and the German Research Foundation through grants SFB568/3, SPP1276 (MetStroem) and KA 3215/1-1. The authors are grateful to the anonymous referee for the valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, USA

    W. Huang & L. Kamenski

  2. Graduate School of Computational Engineering, Center of Smart Interfaces, Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany

    J. Lang

Authors
  1. W. Huang
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  2. L. Kamenski
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  3. J. Lang
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Corresponding author

Correspondence to L. Kamenski .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

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Huang, W., Kamenski, L., Lang, J. (2013). Adaptive Finite Elements with Anisotropic Meshes. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_4

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