High-order Finite Elements on Pyramids: Approximation Spaces, Unisolvency and Exactness

We present a family of high-order finite element approximation spaces on a pyramid, and associated unisolvent degrees of freedom. These spaces consist of rational basis functions. We establish conforming, exactness and polynomial approximation properties.

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

[2]  Leszek Demkowicz,et al.  Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations , 2008 .

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

[4]  Steven J. Owen,et al.  Formation of pyramid elements for hexahedra to tetrahedra transitions , 2001 .

[5]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[6]  L. Demkowicz,et al.  De Rham diagram for hp finite element spaces , 2000 .

[7]  O. Picon,et al.  A finite element method based on Whitney forms to solve Maxwell equations in the time domain , 1995 .

[8]  Spencer J. Sherwin,et al.  Hierarchical hp finite elements in hybrid domains , 1997 .

[9]  E. Wachspress,et al.  A Rational Finite Element Basis , 1975 .

[10]  Leszek Demkowicz,et al.  Construction of H1-conforming hierarchical shape functions for elements of all shapes and transfinite interpolation , 2010 .

[11]  C. K. Lee,et al.  A new automatic adaptive 3D solid mesh generation scheme for thin‐walled structures , 2005 .

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

[13]  Rafael Muñoz-Sola,et al.  Polynomial Liftings on a Tetrahedron and Applications to the h - p Version of the Finite Element Method in Three Dimensions , 1997 .

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

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

[16]  Thomas Rylander,et al.  Stable FEM-FDTD hybrid method for Maxwell's equations , 2000 .

[17]  J. Douglas,et al.  Prismatic mixed finite elements for second order elliptic problems , 1989 .

[18]  Konstantin Lipnikov,et al.  Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes , 2005, SIAM J. Numer. Anal..

[19]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

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

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

[22]  M. Shashkov,et al.  Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆ , 1997 .

[23]  M. Shashkov,et al.  The mimetic finite difference method on polygonal meshes for diffusion-type problems , 2004 .

[24]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[25]  Carlos A. Felippa,et al.  A compendium of FEM integration formulas for symbolic work , 2004 .

[26]  Conforming discretizations on tetrahedrons , pyramids , prisms and hexahedrons , 2007 .

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