Simplicial finite elements in higher dimensions

Jan Brandts; Sergey Korotov; Michal Křížek

Applications of Mathematics (2007)

  • Volume: 52, Issue: 3, page 251-265
  • ISSN: 0862-7940

Abstract

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Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction.

How to cite

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Brandts, Jan, Korotov, Sergey, and Křížek, Michal. "Simplicial finite elements in higher dimensions." Applications of Mathematics 52.3 (2007): 251-265. <http://eudml.org/doc/33287>.

@article{Brandts2007,
abstract = {Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction.},
author = {Brandts, Jan, Korotov, Sergey, Křížek, Michal},
journal = {Applications of Mathematics},
keywords = {$n$-simplex; finite element method; superconvergence; strengthened Cauchy-Schwarz inequality; discrete maximum principle; -simplex; finite element method; superconvergence; strengthened Cauchy-Schwarz inequality; discrete maximum principle; survey paper},
language = {eng},
number = {3},
pages = {251-265},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Simplicial finite elements in higher dimensions},
url = {http://eudml.org/doc/33287},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Brandts, Jan
AU - Korotov, Sergey
AU - Křížek, Michal
TI - Simplicial finite elements in higher dimensions
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 251
EP - 265
AB - Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction.
LA - eng
KW - $n$-simplex; finite element method; superconvergence; strengthened Cauchy-Schwarz inequality; discrete maximum principle; -simplex; finite element method; superconvergence; strengthened Cauchy-Schwarz inequality; discrete maximum principle; survey paper
UR - http://eudml.org/doc/33287
ER -

References

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  1. 10.1023/A:1024058625449, Numer. Algorithms 32 (2003), 185–191. (2003) MR1989366DOI10.1023/A:1024058625449
  2. 10.1002/1099-1506(200104/05)8:3<191::AID-NLA229>3.0.CO;2-7, Numer. Linear Algebra Appl. 8 (2001), 191–205. (2001) MR1817796DOI10.1002/1099-1506(200104/05)8:3<191::AID-NLA229>3.0.CO;2-7
  3. 10.1017/S0962492906210018, Acta Numer. 15 (2006), 1–135. (2006) MR2269741DOI10.1017/S0962492906210018
  4. On multigrid methods of the two-level type, In: Multigrid Methods. Lecture Notes in Mathematics, Vol. 960, W. Hackbusch, U.  Trotenberg (eds.), Springer-Verlag, Berlin, 1982, pp. 352–367. (1982) Zbl0505.65040MR0685778
  5. 10.1023/B:APOM.0000024520.06175.8b, Appl. Math. 49 (2004), 57–72. (2004) MR2032148DOI10.1023/B:APOM.0000024520.06175.8b
  6. 10.1002/nla.340, Numer. Linear Algebra Appl. 10 (2003), 619–637. (2003) MR2030627DOI10.1002/nla.340
  7. 10.1002/nla.350, Numer. Linear Algebra Appl. 11 (2004), 309–326. (2004) MR2057704DOI10.1002/nla.350
  8. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edition, Cambridge University Press, Cambridge, 2001, pp. 309–326. (2001) MR1827293
  9. The strengthened Cauchy-Bunyakowski-Schwarz inequality for n -simplicial linear finite elements, In: Springer Lecture Notes in Computer Science, Vol. 3401, Springer-Verlag, Berlin, 2005, pp. 203–210. (2005) 
  10. Survey of discrete maximum principles for linear elliptic and parabolic problems, In: Proc. Conf. ECCOMAS  2004, P.  Neittaanmäki et al. (eds.), Univ. of Jyväskylä, 2004, pp. 1–19. (2004) 
  11. Dissection of the path-simplex in  ℝ n into n   path-subsimplices, Linear Algebra Appl. 421 (2007), 382–393. (2007) MR2294350
  12. 10.1093/imanum/23.3.489, IMA  J.  Numer. Anal. 23 (2003), 489–505. (2003) MR1987941DOI10.1093/imanum/23.3.489
  13. The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics  15, Springer-Verlag, New York, 1994. (1994) MR1278258
  14. The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
  15. Optimal points of stresses for tetrahedron linear element, Nat. Sci. J.  Xiangtan Univ. 3 (1980), 16–24. (Chinese) (1980) 
  16. COMSOL, Multiphysics Version  3.3  (2006), Sweden, http://www.femlab.com. 
  17. 10.1016/0898-1221(89)90148-X, Comput. Math. Appl. 17 (1989), 59–71. (1989) Zbl0706.51019MR0994189DOI10.1016/0898-1221(89)90148-X
  18. FEMLAB version  2.2  (2002). Multiphysics in Matlab, for use with Matlab, COMSOL, Sweden, http://www.femlab.com. 
  19. Some remarks on finite element analysis of time-dependent field problems, In: Theory Pract. Finite Elem. Struct. Anal, Univ. Tokyo Press, Tokyo, 1973, pp. 91–106. (1973) Zbl0373.65047
  20. 10.1002/num.1690100511, Numer. Methods Partial Differ. Equations 10 (1994), 651–666. (1994) Zbl0807.65112MR1290950DOI10.1002/num.1690100511
  21. 10.1007/s00211-004-0559-0, Numer. Math. 99 (2005), 669–698. (2005) MR2121074DOI10.1007/s00211-004-0559-0
  22. On the diagonal dominance of stiffness matrices in  3D, East-West J.  Numer. Math. 3 (1995), 59–69. (1995) MR1331484
  23. 10.1007/BF00047538, Acta Appl. Math. 9 (1987), 175–198. (1987) MR0900263DOI10.1007/BF00047538
  24. Finite Element Methods: Superconvergence, Post-processing and A  Posteriori Estimates. Proc. Conf. Univ. of Jyväskylä, 1996. Lecture Notes in Pure and Applied Mathematics, Vol. 196, M.  Křížek, P.  Neittaanmäki, and R.  Stenberg (eds.), Marcel Dekker, New York, 1998. (1998) MR1602809
  25. 10.1007/BF01396415, Numer. Math. 35 (1980), 315–341. (1980) DOI10.1007/BF01396415
  26. 10.1007/BF01389668, Numer. Math. 50 (1986), 57–81. (1986) MR0864305DOI10.1007/BF01389668
  27. Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary, Zh. Vychisl. Mat. Mat. Fiz. 9 (1969), 1102–1120. (1969) MR0295599
  28. On the strong maximum principle for some piecewise linear finite element approximate problems of non-positive type, J.  Fac. Sci., Univ. Tokyo, Sect.  IA Math. 29 (1982), 473–491. (1982) Zbl0488.65052MR0672072
  29. 10.1137/S0036142903425082, SIAM J.  Numer. Anal. 42 (2005), 2188–2217. (2005) Zbl1081.65112MR2139244DOI10.1137/S0036142903425082
  30. 10.2514/3.5067, AIAA J. 7 (1969), 178–180. (1969) DOI10.2514/3.5067

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