Interpolation error estimates for mean value coordinates over convex polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.

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

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

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

[4]  Rüdiger Verfürth,et al.  A note on polynomial approximation in Sobolev spaces , 1999 .

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

[6]  Kai Hormann,et al.  Mean value coordinates for arbitrary planar polygons , 2006, TOGS.

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

[8]  Jean-Paul Berrut,et al.  Convergence rates of derivatives of a family of barycentric rational interpolants , 2011 .

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

[10]  Tao Ju,et al.  A general geometric construction of coordinates in a convex simplicial polytope , 2007, Comput. Aided Geom. Des..

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

[12]  Shai Dekel,et al.  The Bramble-Hilbert Lemma for Convex Domains , 2004, SIAM J. Math. Anal..

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

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

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

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

[17]  Christopher Dyken,et al.  Transfinite mean value interpolation , 2009, Comput. Aided Geom. Des..

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

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

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

[21]  Daniel Cohen-Or,et al.  GPU-assisted positive mean value coordinates for mesh deformations , 2007, Symposium on Geometry Processing.

[22]  Raif M. Rustamov,et al.  A versatile framework for shape description , 2010, The Visual Computer.

[23]  Kup-Sze Choi,et al.  Fast Rendering of Diffusion Curves with Triangles , 2012, IEEE Computer Graphics and Applications.

[24]  Long Chen.  FINITE ELEMENT METHOD , 2013 .

[25]  Jirí Kosinka,et al.  Barycentric interpolation and mappings on smooth convex domains , 2010, SPM '10.

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

[27]  Zeev Farbman,et al.  Coordinates for instant image cloning , 2009, ACM Trans. Graph..

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

[29]  Josiah Manson,et al.  Positive Gordon-Wixom coordinates , 2011, Comput. Aided Des..

[30]  Raif M. Rustamov,et al.  Barycentric Coordinates on Surfaces , 2010, Comput. Graph. Forum.

[31]  Kai Hormann,et al.  Barycentric rational interpolation with no poles and high rates of approximation , 2007, Numerische Mathematik.

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

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

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

[35]  Eugene L. Wachspress,et al.  Basis functions for concave polygons , 2008, Comput. Math. Appl..

[36]  Michal Křížek,et al.  On semiregular families of triangulations and linear interpolation , 1991 .

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

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

[39]  Gary L. Miller,et al.  Estimating interpolation error: a combinatorial approach , 1999, SODA '99.

[40]  Alireza Tabarraei,et al.  APPLICATION OF POLYGONAL FINITE ELEMENTS IN LINEAR ELASTICITY , 2006 .