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Spaces of Simplicial Shapes

  • Jon Eivind VatneEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

When discussing shapes of simplices, e.g. in connection with mesh generation and finite element methods, it is important to have a suitable space parametrizing the shapes. In particular, degenerations of different types can appear as boundary components in various ways. For triangles, we will present two natural parametrizing sets that highlight two different types of degenerations, and then combine the properties into a new parametrizing space that allows a good basis for understanding both types of degenerations. The combined space is constructed by the process of blowing up, which in this simple case is introduction of polar coordinates. For tetrahedra, there are many different types of degenerations. In this short paper we will only give one example of what can be achieved by blowing up a natural model, namely to pull apart the two tetrahedral degenerating types known as slivers and caps.

References

  1. 1.
    J. Brandts, S. Korotov, M. Křížek, Dissection of the path-simplex in \(\mathbb {R}\sp n\) into n path-subsimplices. Linear Algebra Appl. 421(2–3), 382–393 (2007)Google Scholar
  2. 2.
    J. Brandts, A. Hannukainen, S. Korotov, M. Křížek, On angle conditions in the finite element method. SEMA J. 56, 81–95 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Brandts, S. Korotov, M. Křížek, A geometric toolbox for tetrahedral finite element partitions, in Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations, ed. by O. Axelsson, J. Karátson (Bentham Science Publishers Ltd., Sharjah, 2011), pp. 101–122Google Scholar
  4. 4.
    J. Brandts, S. Korotov, M. Křížek, Generalization of the Zlámal condition for simplicial finite elements in R d. Appl. Math. 56, 417–424 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Ciarlet, Basic Error Estimates for Elliptic Problems, vol. II. Handbook of Numerical Analysis, II (North-Holland, Amsterdam, 1991), pp. 17–351Google Scholar
  6. 6.
    H. Edelsbrunner, Geometry and Topology for Mesh Generation (Cambridge University Press, Cambridge, 2006)zbMATHGoogle Scholar
  7. 7.
    A. Hannukainen, S. Korotov, M. Křížek, Maximum angle condition for n-dimensional simplicial elements, in Numerical Mathematics and Advanced Applications - ENUMATH 2017, vol. 126, ed. by F.A. Radu, K. Kumar, I. Berre, J.M. Nordbotten, I.S. Pop (Springer, Cham, 2018). https://doi.org/10.1007/978-3-319-96415-7
  8. 8.
    R. Melrose, Real Blow Up. Lecture notes from a course at the session Introduction to Analysis on Singular Spaces (MSRI, 2008). Available from http://math.mit.edu/~rbm/
  9. 9.
    H.M. Nilsen, J.M. Nordbotten, X. Raynaud, Comparison between cell-centered and noda-based discretization schemes for linear elasticity. Comput. Geosci. 22(1), 233–260 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    J.M. Nordbotten, I. Aavatsmark, G.T. Eigestad, Monotonicity of control volume methods. Numer. Math. 106(2), 255–288 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.E. Vatne, Simplices rarely contain their circumcenter in high dimensions. Appl. Math. 62(3), 213–223 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Ženíšek, The convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Appl. Math. 14, 355–377 (1969)zbMATHGoogle Scholar
  13. 13.
    M. Zlámal, On the finite element method. Numer. Math. 12, 394–409 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Western Norway University of Applied SciencesDepartment of Computing, Mathematics and PhysicsBergenNorway

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