Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

Abstract We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

[1]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

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

[3]  S. Christiansen.  A CONSTRUCTION OF SPACES OF COMPATIBLE DIFFERENTIAL FORMS ON CELLULAR COMPLEXES , 2008 .

[4]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[5]  Charles H. Langmuir,et al.  Calculation of phase equilibrium in mineral-melt systems , 1990 .

[6]  Chandrajit L. Bajaj,et al.  Dual formulations of mixed finite element methods with applications , 2010, Comput. Aided Des..

[7]  Douglas N. Arnold,et al.  Geometric decompositions and local bases for spaces of finite element differential forms , 2008, 0806.1255.

[8]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[9]  Josiah Manson,et al.  Moving Least Squares Coordinates , 2010, Comput. Graph. Forum.

[10]  A. Russo,et al.  New perspectives on polygonal and polyhedral finite element methods , 2014 .

[11]  Anders Logg,et al.  The FEniCS Project Version 1.5 , 2015 .

[12]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[13]  Snorre H. Christiansen,et al.  Smoothed projections in finite element exterior calculus , 2007, Math. Comput..

[14]  Glaucio H. Paulino,et al.  On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes , 2014 .

[15]  A. Bossavit.  A uniform rationale for Whitney forms on various supporting shapes , 2010, Math. Comput. Simul..

[16]  Chandrajit L. Bajaj,et al.  Interpolation error estimates for mean value coordinates over convex polygons , 2011, Adv. Comput. Math..

[17]  Mathieu Desbrun,et al.  A geometric construction of coordinates for convex polyhedra using polar duals , 2005, SGP '05.

[18]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[19]  Chandrajit L. Bajaj,et al.  A generalization for stable mixed finite elements , 2010, SPM '10.

[20]  M. Fortin,et al.  Mixed finite elements for second order elliptic problems in three variables , 1987 .

[21]  N. Sukumar,et al.  Conforming polygonal finite elements , 2004 .

[22]  T. Weiland,et al.  Polygonal finite elements , 2006, IEEE Transactions on Magnetics.

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

[24]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[25]  Markus H. Gross,et al.  A Finite Element Method on Convex Polyhedra , 2007, Comput. Graph. Forum.

[26]  Alexander Rand,et al.  Average Interpolation under the Maximum Angle Condition , 2011, SIAM J. Numer. Anal..

[27]  Kai Hormann,et al.  A general construction of barycentric coordinates over convex polygons , 2006, Adv. Comput. Math..

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

[29]  Gianmarco Manzini,et al.  Mimetic finite difference method , 2014, J. Comput. Phys..

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

[31]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[32]  Wenbin Chen,et al.  Minimal degree H(curl) and H(div) conforming finite elements on polytopal meshes , 2015, Math. Comput..

[33]  R. Sibson.  A vector identity for the Dirichlet tessellation , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[35]  Kai Hormann,et al.  Maximum Entropy Coordinates for Arbitrary Polytopes , 2008, Comput. Graph. Forum.

[36]  H. Whitney.  Geometric Integration Theory , 1957 .

[37]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[39]  D. Arnold.  Finite Element Exterior Calculus , 2018 .

[40]  Mark Meyer,et al.  Harmonic coordinates for character articulation , 2007, ACM Trans. Graph..

[41]  N. Sukumar,et al.  Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons , 2013 .

[42]  Mathieu Desbrun,et al.  Barycentric coordinates for convex sets , 2007, Adv. Comput. Math..

[43]  N. Sukumar.  Construction of polygonal interpolants: a maximum entropy approach , 2004 .

[44]  Markus H. Gross,et al.  Polyhedral Finite Elements Using Harmonic Basis Functions , 2008, Comput. Graph. Forum.

[45]  Joe D. Warren,et al.  Barycentric coordinates for convex polytopes , 1996, Adv. Comput. Math..

[46]  L. D. Marini,et al.  VIRTUAL ELEMENT METHOD FOR PLATE BENDING PROBLEMS , 2012 .

[47]  Eugene L. Wachspress,et al.  Barycentric coordinates for polytopes , 2011, Comput. Math. Appl..

[48]  P. Milbradt,et al.  Polytope finite elements , 2008 .

[49]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .

[50]  Pavel B. Bochev,et al.  Principles of Mimetic Discretizations of Differential Operators , 2006 .

[51]  Chandrajit L. Bajaj,et al.  Error estimates for generalized barycentric interpolation , 2010, Adv. Comput. Math..

[52]  A. F. Rasmussen,et al.  Velocity interpolation and streamline tracing on irregular geometries , 2012, Computational Geosciences.

[53]  M. M. Rashid,et al.  A three‐dimensional finite element method with arbitrary polyhedral elements , 2006 .

[54]  Michael S. Floater,et al.  Mean value coordinates , 2003, Comput. Aided Geom. Des..

[55]  Andrew Kruse Gillette.  Stability of dual discretization methods for partial differential equations , 2011 .

[56]  Chandrajit L. Bajaj,et al.  Quadratic serendipity finite elements on polygons using generalized barycentric coordinates , 2011, Math. Comput..

[57]  Michael S. Floater,et al.  Gradient Bounds for Wachspress Coordinates on Polytopes , 2013, SIAM J. Numer. Anal..

[58]  Wolfgang Knapp,et al.  Whitney Elements on Sparse Grids , 2002 .

[59]  Lourenço Beirão da Veiga,et al.  Virtual Elements for Linear Elasticity Problems , 2013, SIAM J. Numer. Anal..

[60]  Gerald E. Farin,et al.  Surfaces over Dirichlet tessellations , 1990, Comput. Aided Geom. Des..

[61]  Gianmarco Manzini,et al.  Mimetic scalar products of discrete differential forms , 2014, J. Comput. Phys..

[62]  Martin Reimers,et al.  Mean value coordinates in 3D , 2005, Comput. Aided Geom. Des..

[63]  A. Bossavit.  Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism , 1988 .

[64]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .