Numerical integration for high order pyramidal finite elements
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 2, pp. 239-263.

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.

DOI : https://doi.org/10.1051/m2an/2011042
Classification : 65N30,  65D30
Mots clés : finite elements, quadrature, pyramid
@article{M2AN_2012__46_2_239_0,
author = {Nigam, Nilima and Phillips, Joel},
title = {Numerical integration for high order pyramidal finite elements},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {239--263},
publisher = {EDP-Sciences},
volume = {46},
number = {2},
year = {2012},
doi = {10.1051/m2an/2011042},
zbl = {1276.65083},
mrnumber = {2855642},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011042/}
}
Nigam, Nilima; Phillips, Joel. Numerical integration for high order pyramidal finite elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 2, pp. 239-263. doi : 10.1051/m2an/2011042. http://www.numdam.org/articles/10.1051/m2an/2011042/

[1] D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Num. 15 (2006) 1-155. | MR 2269741 | Zbl 1185.65204

[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. | MR 2594630 | Zbl 1207.65134

[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. | MR 2585588 | Zbl 1203.65243

[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. | MR 263214 | Zbl 0201.07803

[5] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer Verlag (2008). | MR 2373954 | Zbl 0804.65101

[6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Society for Industrial Mathematics (2002). | MR 1930132 | Zbl 0383.65058

[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. | MR 2105164 | Zbl 1143.78365

[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). | Zbl 1148.65001

[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). | MR 1115205 | Zbl 0788.73002

[11] V. Gradinaru and R. Hiptmair, Whitney elements on pyramids. Electronic Transactions on Numerical Analysis 8 (1999) 154-168. | MR 1744532 | Zbl 0970.65120

[12] R.D. Graglia and I.L. Gheorma, Higher order interpolatory vector bases on pyramidal elements. IEEE Trans. Antennas Propag. 47 (1999) 775. | Zbl 0945.78016

[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. | MR 86389 | Zbl 0070.35404

[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. | Zbl 0985.65141

[15] P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003). | MR 2059447 | Zbl 1024.78009

[16] J.-C. Nedéléc, Mixed finite elements in ${ℝ}^{3}$. Num. Math. 35 (1980) 315-341. | Zbl 0419.65069

[17] N. Nigam and J. Phillips, High-order conforming finite elements on pyramids. IMA J. Numer. Anal. (2011); doi: 10.1093/imanum/drr015. | MR 2911396 | Zbl 1241.65102

[18] A.H. Stroud, Approximate calculation of multiple integrals. Prentice-Hall Inc., Englewood Cliffs, N.J. (1971). | MR 327006 | Zbl 0379.65013

[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. | MR 2039968 | Zbl 1043.52009

[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.