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 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 }, pages = {239--263}, publisher = {EDP-Sciences}, volume = {46}, number = {2}, year = {2012}, doi = {10.1051/m2an/2011042}, mrnumber = {2855642}, zbl = {1276.65083}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2011042/} }
TY - JOUR AU - Nigam, Nilima AU - Phillips, Joel TI - Numerical integration for high order pyramidal finite elements JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 239 EP - 263 VL - 46 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011042/ DO - 10.1051/m2an/2011042 LA - en ID - M2AN_2012__46_2_239_0 ER -
%0 Journal Article %A Nigam, Nilima %A Phillips, Joel %T Numerical integration for high order pyramidal finite elements %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 239-263 %V 46 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011042/ %R 10.1051/m2an/2011042 %G en %F M2AN_2012__46_2_239_0
Nigam, Nilima; Phillips, Joel. Numerical integration for high order pyramidal finite elements. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 (2012) no. 2, pp. 239-263. doi : 10.1051/m2an/2011042. http://www.numdam.org/articles/10.1051/m2an/2011042/
[1] Finite element exterior calculus, homological techniques, and applications. Acta Num. 15 (2006) 1-155. | MR | Zbl
, and ,[2] Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47 (2010) 281-354. | MR | Zbl
, and ,[3] Higher-order finite elements for hybrid meshes using new nodal pyramidal elements. J. Sci. Comput. 42 (2010) 345-381. | MR | Zbl
, and ,[4] Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112-124. | MR | Zbl
and ,[5] The mathematical theory of finite element methods. Springer Verlag (2008). | MR | Zbl
and ,[6] The Finite Element Method for Elliptic Problems. Society for Industrial Mathematics (2002). | MR | Zbl
,[7] A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements. IEEE Trans. Magn. 33 (1997) 1362-1365.
, and ,[8] H1, and -conforming projection-based interpolation in three dimensions. Quasi-optimal -interpolation estimates. Comput. Methods Appl. Mech. Eng. 194 (2005) 267-296. | MR | Zbl
and ,[9] Computing with hp-Adaptive Finite Elements Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications 2. Chapman & Hall (2007). | Zbl
, , , and ,[10] Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics). Springer-Verlag Berlin and Heidelberg GmbH & Co. K (1991). | MR | Zbl
and ,[11] Whitney elements on pyramids. Electronic Transactions on Numerical Analysis 8 (1999) 154-168. | MR | Zbl
and ,[12] Higher order interpolatory vector bases on pyramidal elements. IEEE Trans. Antennas Propag. 47 (1999) 775. | Zbl
and ,[13] Numerical integration over simplexes and cones. Mathematical Tables Aids Comput. 10 (1956) 130-137. | MR | Zbl
, and ,[14] Fully discrete hp-finite elements: Fast quadrature. Comput. Methods Appl. Mech. Eng. 190 (2001) 4339-4364. | Zbl
, and ,[15] Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003). | MR | Zbl
,[16] Mixed finite elements in . Num. Math. 35 (1980) 315-341. | Zbl
,[17] High-order conforming finite elements on pyramids. IMA J. Numer. Anal. (2011); doi: 10.1093/imanum/drr015. | MR | Zbl
and ,[18] Approximate calculation of multiple integrals. Prentice-Hall Inc., Englewood Cliffs, N.J. (1971). | MR | Zbl
,[19] 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 | Zbl
,[20] Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons. Technical report, University of Stuttgart.
,[21] Finite Element methods for Electromagnetic Field Computation. Ph. D. thesis, Johannes Kepler University, Linz (2006).
,[22] A new family of finite elements: the pyramidal elements. IEEE Trans. Magn. 32 (1996) 1393-1396.
, , , and ,Cited by Sources: