Numerical integration for high order pyramidal finite elements

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.

[1]  Ralf Hiptmair,et al.  Whitney elements on pyramids. , 1999 .

[2]  J. Bramble,et al.  Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation , 1970 .

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  Nilima Nigam,et al.  High-order Finite Elements on Pyramids: Approximation Spaces, Unisolvency and Exactness , 2006, math/0610206.

[5]  J.-L. Coulomb,et al.  A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements , 1997 .

[6]  N. Nigam,et al.  Higher-order finite elements on pyramids , 2006 .

[7]  Leszek Demkowicz,et al.  H1, H(curl) and H(div)-conforming projection-based interpolation in three dimensionsQuasi-optimal p-interpolation estimates , 2005 .

[8]  Jens Markus Melenk,et al.  Fully discrete hp-finite elements: fast quadrature , 2001 .

[9]  J. Warren.  On the Uniqueness of Barycentric Coordinates , 2003 .

[10]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[11]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[12]  Marc Duruflé,et al.  Higher-order Finite Elements for Hybrid Meshes Using New Nodal Pyramidal Elements , 2010, J. Sci. Comput..

[13]  N. Nigam,et al.  High-order finite elements on pyramids. II: unisolvency and exactness , 2010, 1006.5240.

[14]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[15]  Frank Claeyssen,et al.  A new family of finite elements: the pyramidal elements , 1996 .

[16]  Andrew F. Peterson,et al.  Higher order interpolatory vector bases on pyramidal elements , 1998 .

[17]  Maciej Paszyński,et al.  Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications , 2007 .

[18]  D. P. Flemming.  Numerical Integration over Simplexes and Cones , 2010 .

[19]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[20]  G. Fix.  Review: Philippe G. Ciarlet, The finite element method for elliptic problems , 1979 .

[21]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[22]  Nilima Nigam,et al.  High-order conforming finite elements on pyramids , 2012 .

[23]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[24]  Franco Brezzi Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics) , 1991 .