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Nonobtuse Tetrahedral Partitions that Refine Locally Towards Fichera-Like Corners

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Abstract

Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain.

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References

  1. T. Apel: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numer. Math. B. G. Teubner, Leipzig, 1999.

    Google Scholar 

  2. T. Apel, F. Milde: Comparison of several mesh refinement strategies near edges. Commun. Numer. Methods Eng. 12 (1996), 373–381.

    MATH  Google Scholar 

  3. T. Apel, S. Nicaise: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21 (1998), 519–549.

    Article  MathSciNet  MATH  Google Scholar 

  4. O. Axelsson, V. A. Barker: Finite Element Solution of Boundary Value Problems. Theory and Computation. Academic Press, Orlando, 1984.

    Google Scholar 

  5. M. Bern, P. Chew, D. Eppstein, and J. Ruppert: Dihedral bounds for mesh generation in high dimensions. Proc. of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1995). SIAM, Philadelphia, 1995, pp. 189–196.

    Google Scholar 

  6. F. Bornemann, B. Erdmann, and R. Kornhuber: Adaptive multilevel methods in three space dimensions. Int. J. Numer. Methods Eng. 36 (1993), 3187–3203.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Bansch: Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Eng. 3 (1991), 181–191.

    MathSciNet  Google Scholar 

  8. L. Collatz: Numerische Behandlung von Differentialgleichungen. Springer-Verlag, Berlin-Gottingen-Heidelberg, 1951.

    MATH  Google Scholar 

  9. R. Cools: Monomial cubature rules since “Stroud”: A compilation. II. J. Comput. Appl. Math. 112 (1999), 21–27.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Cools, P. Rabinowitz: Monomial cubature rules since “Stroud”: A compilation. J. Comput. Appl. Math. 48 (1993), 309–326.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. S. M. Coxeter: Trisecting an orthoscheme. Comput. Math. Appl. 17 (1989), 59–71.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Dauge: Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math., Vol. 1341. Springer-Verlag, Berlin, 1988.

  13. M. Feistauer, J. Felcman, M. Rokyta, and Z. Vlasek: Finite-element solution of flow problems with trailing conditions. J. Comput. Appl. Math. 44 (1992), 131–165.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Fichera: Numerical and Quantitative Analysis. Surveys and Reference Works in Mathematics, Vol. 3. Pitman (Advanced Publishing Program), London-San Francisco-Melbourne, 1978.

    Google Scholar 

  15. H. Fujii: Some remarks on finite element analysis of time-dependent field problems. Theory Pract. Finite Elem. Struct. Analysis. Univ. Tokyo Press, Tokyo, 1973, pp. 91–106.

    Google Scholar 

  16. B. Q. Guo: The h-p version of the finite element method for solving boundary value problems in polyhedral domains. Boundary Value Problems and Integral Equations in Nonsmooth Domains (Luminy, 1993). Lect. Notes Pure Appl. Math., Vol. 167 (M. Costabel, M. Dauge, and C. Nicaise, eds.). Marcel Dekker, New York, 1995, pp. 101–120.

    Google Scholar 

  17. P. Keast: Moderate-degree tetrahedral quadrature formulas. Comput. Methods Appl. Mech. Eng. 55 (1986), 339–348.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Korotov, M. Krizek: Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39 (2001), 724–733.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Korotov, M. Krizek: Local nonobtuse tetrahedral refinements of a cube. Appl. Math. Lett. 16 (2003), 1101–1104.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Kossaczky: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55 (1994), 275–288.

    MATH  MathSciNet  Google Scholar 

  21. M. Krizek, L. Liu: On the maximum and comparison principles for a steady-state non-linear heat conduction problem. Z. Angew. Math. Mech. 83 (2003), 559–563.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Krizek, Qun Lin: On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 3 (1995), 59–69.

    MathSciNet  MATH  Google Scholar 

  23. M. Krizek, T. Strouboulis: How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition. Numer. Methods Partial Differ. Equations 13 (1997), 201–214.

    MathSciNet  MATH  Google Scholar 

  24. J. M. Maubach: Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16 (1995), 210–227.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Picasso: Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz-Zhu error estimator. Commun. Numer. Methods Eng. 19 (2003), 13–23.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Plaza, G. F. Carey: Local refinement of simplicial grids based on the skeleton. Appl. Numer. Math. 32 (2000), 195–218.

    MathSciNet  MATH  Google Scholar 

  28. H. Schmitz, K. Volk, and W. Wendland: Three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differ. Equations 9 (1993), 323–337.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. S. Varga: Matrix iterative analysis. Prentice-Hall, New Jersey, 1962.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Basel, Rheinsprung 21, CH-4051, Basel, Switzerland

    Larisa Beilina

  2. Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FI-02015, Espoo, Finland

    Sergey Korotov

  3. Mathematical Institute, Academy of Sciences of the Czech Republic, Zitna 25, CZ-115 67, Praha 1, Czech Republic

    Michal Krizek

Authors
  1. Larisa Beilina
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  2. Sergey Korotov
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  3. Michal Krizek
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Additional information

The first author was supported by the Swedish Foundation for Strategic Research, the second author was supported by Grant No. 49051 of the Academy of Finland, the third author was supported by Grant No. A 1019201 of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AV0Z 10190503.

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Beilina, L., Korotov, S. & Krizek, M. Nonobtuse Tetrahedral Partitions that Refine Locally Towards Fichera-Like Corners. Appl Math 50, 569–581 (2005). https://doi.org/10.1007/s10492-005-0038-7

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  • DOI: https://doi.org/10.1007/s10492-005-0038-7

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