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Proceedings of the 20th International Meshing Roundtable pp 313–327Cite as

Families of Meshes Minimizing P 1 Interpolation Error

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Summary

For a given function, we consider a problem of minimizing the P 1 interpolation error on a set of triangulations with a fixed number of triangles. The minimization problem is reformulated as a problem of generating a mesh which is quasi-uniform in a specially designed metric. For functions with indefinite Hessian, we show existence of a family of metrics with highly diverse properties. The family may include both anisotropic and isotropic metrics. A developed theory is verified with numerical examples.

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References

  1. Agouzal, A., Lipnikov, K., Vassilevski, Y.: Adaptive generation of quasi-optimal tetrahedral meshes. East-West J. Numer. Math. 7, 223–244 (1999)

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Agouzal, A., Vassilevski, Y.: Minimization of gradient errors of piecewise linear interpolation on simplicial meshes. Comp. Meth. Appl. Mech. Engnr. 199, 2195–2203 (2010)

    Article  MathSciNet  Google Scholar 

  4. Agouzal, A., Lipnikov, K., Vassilevski, Y.: Edge-based a posteriori error estimators for generating quasi-optimal simplicial meshes. Math. Model. Nat. Phenom. 5, 91–96 (2010)

    Article  MathSciNet  Google Scholar 

  5. Agouzal, A., Lipnikov, K., Vassilevski, Y.: On optimal convergence rate of finite element solutions of boundary value problems on adaptive anisotropic meshes. Math. Comput. Simul. (in Press, 2011)

    Google Scholar 

  6. Borouchaki, H., Hecht, F., Frey, P.J.: Mesh gradation control. Inter. J. Numer. Meth. Engrg. 43, 1143–1165 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Buscaglia, G.C., Dari, D.A.: Anisotropic mesh optimization and its application in adaptivity. Int. J. Numer. Meth. Eng. 40, 4119–4136 (1997)

    Article  MATH  Google Scholar 

  8. Chen, L., Sun, P., Xu, J.: Optimal anisotropic meshes for minimizing interpolation errors in L p-norm. Mathematics of Computation 76, 179–204 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland (1978)

    Google Scholar 

  10. Coupez, T.: Generation de maillage et adaptation de maillage par optimisation locale. Revue Europeenne Des Elements Finis 49, 403–423 (2000) (in French)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Farrell, P.E., Maddison, J.R.: Conservative interpolation between volume meshes by local Galerkin projection. Comput. Methods Appl. Mech. Engrg. 200, 89–100 (2011)

    Article  MathSciNet  Google Scholar 

  13. Fortin, M., Vallet, M.-G., Dompierre, J., Bourgault, Y., Habashi, W.G.: Anisotropic mesh adaptation: theory, validation and applications Computational Fluid dynamics, pp. 174–180. John Wiley & Sons Ltd (1996)

    Google Scholar 

  14. George, P.L.: Automatic mesh generation: Applications to Finite Element Methods. John Wiley & Sons, Inc., New York (1991)

    MATH  Google Scholar 

  15. Huang, W., Sun, W.: Variational mesh adaptation II: Error estimates and monitor functions. J. Comput. Phys. 184, 619–648 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Loseille, A., Alauzet, F.: Optimal 3D highly anisotropic mesh adaptation based on the continuous mesh framework. In: Clark, B.W. (ed.) Proc. of 18th Int. Meshing Roundtable, pp. 575–594 (2009)

    Google Scholar 

  18. Vassilevski, Y., Lipnikov, K.: Adaptive algorithm for generation of quasi-optimal meshes. Comp. Math. Math. Phys. 39, 1532–1551 (1999)

    Google Scholar 

  19. Vassilevski, Y., Agouzal, A.: An unified asymptotic analysis of interpolation errors for optimal meshes. Doklady Mathematics 72, 879–882 (2005)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. I.C.J. Universite Lyon1, France

    A. Agouzal

  2. Los Alamos National Laboratory, Los Alamos, NM, 87545, U.S.A.

    K. Lipnikov

  3. Institute of Numerical Mathematics, Gubkina str. 8, Moscow, 119333, Russia

    Y. Vassilevski

Authors
  1. A. Agouzal
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  2. K. Lipnikov
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  3. Y. Vassilevski
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Editor information

Editors and Affiliations

  1. Sandia National Laboratories, P.O. Box 5800, 87185, Albuquerque, NM, USA

    William Roshan Quadros

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© 2011 Springer-Verlag Berlin Heidelberg

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Agouzal, A., Lipnikov, K., Vassilevski, Y. (2011). Families of Meshes Minimizing P 1 Interpolation Error. In: Quadros, W.R. (eds) Proceedings of the 20th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24734-7_17

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  • DOI: https://doi.org/10.1007/978-3-642-24734-7_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24733-0

  • Online ISBN: 978-3-642-24734-7

  • eBook Packages: EngineeringEngineering (R0)

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