# Numerical integration for high order pyramidal finite elements

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

## Access Full Article

top## Abstract

top## How to cite

topNigam, 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

top- [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] 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] 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] 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] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer Verlag (2008). Zbl0804.65101MR2373954
- [6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Society for Industrial Mathematics (2002). Zbl0383.65058MR1930132
- [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] L. Demkowicz and A. Buffa, H1, $H\left(\mathrm{curl}\right)$ and $H\left(\mathrm{div}\right)$-conforming projection-based interpolation in three dimensions. Quasi-optimal $p$-interpolation estimates. Comput. Methods Appl. Mech. Eng. 194 (2005) 267–296. Zbl1143.78365MR2105164
- [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] 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] V. Gradinaru and R. Hiptmair, Whitney elements on pyramids. Electronic Transactions on Numerical Analysis8 (1999) 154–168. Zbl0970.65120MR1744532
- [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] 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] 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] P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003). Zbl1024.78009MR2059447
- [16] J.-C. Nedéléc, Mixed finite elements in ${\mathbb{R}}^{3}$. Num. Math.35 (1980) 315–341. Zbl0419.65069
- [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] A.H. Stroud, Approximate calculation of multiple integrals. Prentice-Hall Inc., Englewood Cliffs, N.J. (1971). Zbl0379.65013MR327006
- [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] C. Wieners, Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons. Technical report, University of Stuttgart.
- [21] S. Zaglmayr, High Order Finite Element methods for Electromagnetic Field Computation. Ph. D. thesis, Johannes Kepler University, Linz (2006).
- [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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.