Numerical integration for high order pyramidal finite elements

Nilima Nigam; Joel Phillips

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 2, page 239-263
  • ISSN: 0764-583X

Abstract

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We examine the effect of numerical integration on the accuracy of high order conforming pyramidal finite element methods. Non-smooth shape functions are indispensable to the construction of pyramidal elements, and this means the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include non-smooth functions and show that, despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.

How to cite

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Nigam, Nilima, and Phillips, Joel. "Numerical integration for high order pyramidal finite elements." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 239-263. <http://eudml.org/doc/273205>.

@article{Nigam2012,
abstract = {We examine the effect of numerical integration on the accuracy of high order conforming pyramidal finite element methods. Non-smooth shape functions are indispensable to the construction of pyramidal elements, and this means the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include non-smooth functions and show that, despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.},
author = {Nigam, Nilima, Phillips, Joel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite elements; quadrature; pyramid; higher-order conforming pyramidal finite element methods; quadrature rule; convergence; consistency; discrete de Rham complex},
language = {eng},
number = {2},
pages = {239-263},
publisher = {EDP-Sciences},
title = {Numerical integration for high order pyramidal finite elements},
url = {http://eudml.org/doc/273205},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Nigam, Nilima
AU - Phillips, Joel
TI - Numerical integration for high order pyramidal finite elements
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 2
SP - 239
EP - 263
AB - We examine the effect of numerical integration on the accuracy of high order conforming pyramidal finite element methods. Non-smooth shape functions are indispensable to the construction of pyramidal elements, and this means the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include non-smooth functions and show that, despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.
LA - eng
KW - finite elements; quadrature; pyramid; higher-order conforming pyramidal finite element methods; quadrature rule; convergence; consistency; discrete de Rham complex
UR - http://eudml.org/doc/273205
ER -

References

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  1. [1] D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Num.15 (2006) 1–155. Zbl1185.65204MR2269741
  2. [2] D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc.47 (2010) 281–354. Zbl1207.65134MR2594630
  3. [3] M. Bergot, G. Cohen and M. Duruflé, Higher-order finite elements for hybrid meshes using new nodal pyramidal elements. J. Sci. Comput.42 (2010) 345–381. Zbl1203.65243MR2585588
  4. [4] J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7 (1970) 112–124. Zbl0201.07803MR263214
  5. [5] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer Verlag (2008). Zbl0804.65101MR2373954
  6. [6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Society for Industrial Mathematics (2002). Zbl0383.65058MR1930132
  7. [7] J.L. Coulomb, F.X. Zgainski and Y. Maréchal, A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements. IEEE Trans. Magn.33 (1997) 1362–1365. 
  8. [8] L. Demkowicz and A. Buffa, H1, H ( curl ) and H ( div ) -conforming projection-based interpolation in three dimensions. Quasi-optimal p -interpolation estimates. Comput. Methods Appl. Mech. Eng. 194 (2005) 267–296. Zbl1143.78365MR2105164
  9. [9] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski and W. Rachowicz, Computing with hp-Adaptive Finite Elements Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications 2. Chapman & Hall (2007). Zbl1148.65001
  10. [10] M. Fortin and F. Brezzi, Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics). Springer-Verlag Berlin and Heidelberg GmbH & Co. K (1991). Zbl0788.73002MR1115205
  11. [11] V. Gradinaru and R. Hiptmair, Whitney elements on pyramids. Electronic Transactions on Numerical Analysis8 (1999) 154–168. Zbl0970.65120MR1744532
  12. [12] R.D. Graglia and I.L. Gheorma, Higher order interpolatory vector bases on pyramidal elements. IEEE Trans. Antennas Propag. 47 (1999) 775. Zbl0945.78016
  13. [13] P.C. Hammer, O.J. Marlowe and A.H. Stroud, Numerical integration over simplexes and cones. Mathematical Tables Aids Comput.10 (1956) 130–137. Zbl0070.35404MR86389
  14. [14] J.M. Melenk, K. Gerdes and C. Schwab, Fully discrete hp-finite elements: Fast quadrature. Comput. Methods Appl. Mech. Eng.190 (2001) 4339–4364. Zbl0985.65141
  15. [15] P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003). Zbl1024.78009MR2059447
  16. [16] J.-C. Nedéléc, Mixed finite elements in 3 . Num. Math.35 (1980) 315–341. Zbl0419.65069
  17. [17] N. Nigam and J. Phillips, High-order conforming finite elements on pyramids. IMA J. Numer. Anal. (2011); doi: 10.1093/imanum/drr015. Zbl1241.65102MR2911396
  18. [18] A.H. Stroud, Approximate calculation of multiple integrals. Prentice-Hall Inc., Englewood Cliffs, N.J. (1971). Zbl0379.65013MR327006
  19. [19] J. Warren, On the uniqueness of barycentric coordinates, in Topics in Algebraic Geometry and Geometric Modeling: Workshop on Algebraic Geometry and Geometric Modeling, July 29-August 2, 2002, Vilnius University, Lithuania. American Mathematical Society 334 (2002) 93–99. Zbl1043.52009MR2039968
  20. [20] C. Wieners, Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons. Technical report, University of Stuttgart. 
  21. [21] S. Zaglmayr, High Order Finite Element methods for Electromagnetic Field Computation. Ph. D. thesis, Johannes Kepler University, Linz (2006). 
  22. [22] F.-X. Zgainski, J.-L. Coulomb, Y. Marechal, F. Claeyssen and X. Brunotte, A new family of finite elements: the pyramidal elements. IEEE Trans. Magn.32 (1996) 1393–1396. 

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