Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-17T02:01:57.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical integration for high order pyramidal finite elements

Published online by Cambridge University Press:  12 October 2011

Nilima Nigam  and
Joel Phillips
Nilima Nigam
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada. nigam@math.sfu.ca
Joel Phillips
Affiliation:
Department of Mathematics, University College London, Main Campus Gower Street Bloombury, WC1E 6BT, London, UK. joel.phillips@ucl.ac.uk
Get access

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.

Keywords

Finite elementsquadraturepyramid
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 239 - 263
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N., Falk, R.S. and Winther, R., Finite element exterior calculus, homological techniques, and applications. Acta Num. 15 (2006) 1–155. CrossRef
Arnold, D.N., Falk, R.S. and Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47 (2010) 281–354. CrossRef
Bergot, M., Cohen, G. and Duruflé, M., Higher-order finite elements for hybrid meshes using new nodal pyramidal elements. J. Sci. Comput. 42 (2010) 345–381. CrossRef
Bramble, J.H. and Hilbert, S.R., Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112–124. CrossRef
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer Verlag (2008).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Society for Industrial Mathematics (2002).
Coulomb, J.L., Zgainski, F.X. and Maréchal, Y., A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements. IEEE Trans. Magn. 33 (1997) 1362–1365. CrossRef
Demkowicz, L. and Buffa, A., H 1, $H({\rm curl})$ and $H({\rm div})$-conforming projection-based interpolation in three dimensions. Quasi-optimal $p$-interpolation estimates. Comput. Methods Appl. Mech. Eng. 194 (2005) 267–296.
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).
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).
Gradinaru, V. and Hiptmair, R., Whitney elements on pyramids. Electronic Transactions on Numerical Analysis 8 (1999) 154–168.
Graglia, R.D. and Gheorma, I.L., Higher order interpolatory vector bases on pyramidal elements. IEEE Trans. Antennas Propag. 47 (1999) 775. CrossRef
Hammer, P.C., Marlowe, O.J. and Stroud, A.H., Numerical integration over simplexes and cones. Mathematical Tables Aids Comput. 10 (1956) 130–137.
Melenk, J.M., Gerdes, K. and Schwab, C., Fully discrete hp-finite elements: Fast quadrature. Comput. Methods Appl. Mech. Eng. 190 (2001) 4339–4364. CrossRef
P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003).
Nedéléc, J.-C., Mixed finite elements in $\mathbb{R}^3$. Num. Math. 35 (1980) 315–341. CrossRef
N. Nigam and J. Phillips, High-order conforming finite elements on pyramids. IMA J. Numer. Anal. (2011); doi: 10.1093/imanum/drr015.
A.H. Stroud, Approximate calculation of multiple integrals. Prentice-Hall Inc., Englewood Cliffs, N.J. (1971).
Warren, J., 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.
C. Wieners, Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons. Technical report, University of Stuttgart.
S. Zaglmayr, High Order Finite Element methods for Electromagnetic Field Computation. Ph. D. thesis, Johannes Kepler University, Linz (2006).
Zgainski, F.-X., Coulomb, J.-L., Marechal, Y., Claeyssen, F. and Brunotte, X., A new family of finite elements: the pyramidal elements. IEEE Trans. Magn. 32 (1996) 1393–1396. CrossRef