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Quadratic serendipity finite elements on polygons using generalized barycentric coordinates


Authors: Alexander Rand, Andrew Gillette and Chandrajit Bajaj
Journal: Math. Comp. 83 (2014), 2691-2716
MSC (2010): Primary 65D05, 65N30, 41A30, 41A25
DOI: https://doi.org/10.1090/S0025-5718-2014-02807-X
Published electronically: February 20, 2014
MathSciNet review: 3246806
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Abstract: We introduce a finite element construction for use on the class of convex, planar polygons and show that it obtains a quadratic error convergence estimate. On a convex $ n$-gon, our construction produces $ 2n$ basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of $ n(n+1)/2$ basis functions known to obtain quadratic convergence. This technique broadens the scope of the so-called `serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.


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  • [1] D. Apprato, R. Arcangéli, and J. L. Gout, Rational interpolation of Wachspress error estimates, Comput. Math. Appl. 5 (1979), no. 4, 329-336. MR 551405 (80m:65011), https://doi.org/10.1016/0898-1221(79)90092-0
  • [2] D. Apprato, R. Arcangéli, and J. L. Gout, Sur les éléments finis rationnels de Wachspress, Numer. Math. 32 (1979), no. 3, 247-270 (French, with English summary). MR 535193 (81h:65109), https://doi.org/10.1007/BF01397000
  • [3] Douglas N. Arnold and Gerard Awanou, The serendipity family of finite elements, Found. Comput. Math. 11 (2011), no. 3, 337-344. MR 2794906 (2012i:65249), https://doi.org/10.1007/s10208-011-9087-3
  • [4] Douglas N. Arnold, Daniele Boffi, and Richard S. Falk, Approximation by quadrilateral finite elements, Math. Comp. 71 (2002), no. 239, 909-922 (electronic). MR 1898739 (2003c:65112), https://doi.org/10.1090/S0025-5718-02-01439-4
  • [5] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954 (2008m:65001)
  • [6] D. Chavey, Tilings by regular polygons. II. A catalog of tilings, Comput. Math. Appl. 17 (1989), no. 1-3, 147-165. Symmetry 2: unifying human understanding, Part 1. MR 994197 (90f:52020), https://doi.org/10.1016/0898-1221(89)90156-9
  • [7] Snorre H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 739-757. MR 2413036 (2009a:58001), https://doi.org/10.1142/S021820250800284X
  • [8] Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132
  • [9] E. Cueto, N. Sukumar, B. Calvo, M. A. Martínez, J. Cegoñino, and M. Doblaré, Overview and recent advances in natural neighbour Galerkin methods, Arch. Comput. Methods Engrg. 10 (2003), no. 4, 307-384. MR 2032470 (2004m:65190), https://doi.org/10.1007/BF02736253
  • [10] S. Dekel and D. Leviatan, The Bramble-Hilbert lemma for convex domains, SIAM J. Math. Anal. 35 (2004), no. 5, 1203-1212. MR 2050198 (2005d:41010), https://doi.org/10.1137/S0036141002417589
  • [11] Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138 (2005d:65002)
  • [12] Gerald Farin, Surfaces over Dirichlet tessellations, Comput. Aided Geom. Design 7 (1990), no. 1-4, 281-292. Curves and surfaces in CAGD '89 (Oberwolfach, 1989). MR 1074614 (91g:65037), https://doi.org/10.1016/0167-8396(90)90036-Q
  • [13] Michael S. Floater, Mean value coordinates, Comput. Aided Geom. Design 20 (2003), no. 1, 19-27. MR 1968304, https://doi.org/10.1016/S0167-8396(03)00002-5
  • [14] M. Floater, K. Hormann, and G. Kós, A general construction of barycentric coordinates over convex polygons, Adv. Comput. Math. 24 (2006), no. 1, 311-331.
  • [15] A. Gillette and C. Bajaj, A generalization for stable mixed finite elements, Proc. 14th ACM Symp. Solid Phys. Modeling, 2010, pp. 41-50.
  • [16] -, Dual formulations of mixed finite element methods with applications, Comput. Aided Des. 43 (2011), no. 10, 1213-1221.
  • [17] Andrew Gillette, Alexander Rand, and Chandrajit Bajaj, Error estimates for generalized barycentric interpolation, Adv. Comput. Math. 37 (2012), no. 3, 417-439. MR 2970859, https://doi.org/10.1007/s10444-011-9218-z
  • [18] J. L. Gout, Construction of a Hermite rational ``Wachspress-type'' finite element, Comput. Math. Appl. 5 (1979), no. 4, 337-347. MR 551406 (81d:65064), https://doi.org/10.1016/0898-1221(79)90093-2
  • [19] J.-L. Gout, Rational Wachspress-type finite elements on regular hexagons, IMA J. Numer. Anal. 5 (1985), no. 1, 59-77. MR 777959 (86g:65210), https://doi.org/10.1093/imanum/5.1.59
  • [20] Thomas J. R. Hughes, The finite element method, Prentice Hall Inc., Englewood Cliffs, NJ, 1987. Linear static and dynamic finite element analysis; With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. MR 1008473 (90i:65001)
  • [21] P. Joshi, M. Meyer, T. DeRose, B. Green, and T. Sanocki, Harmonic coordinates for character articulation, ACM Trans. Graphics 26 (2007), 71.
  • [22] F. Kikuchi, M. Okabe, and H. Fujio, Modification of the 8-node serendipity element, Comput. Methods Appl. Mech. Engrg. 179 (1999), no. 1-2, 91-109.
  • [23] T. Langer and H.P. Seidel, Higher order barycentric coordinates, Comput. Graphics Forum, vol. 27, Wiley Online Library, 2008, pp. 459-466.
  • [24] R. H. MacNeal and R. L. Harder, Eight nodes or nine?, Int. J. Numer. Methods Eng. 33 (1992), no. 5, 1049-1058.
  • [25] S. Martin, P. Kaufmann, M. Botsch, M. Wicke, and M. Gross, Polyhedral finite elements using harmonic basis functions, Proc. Symp. Geom. Proc., 2008, pp. 1521-1529.
  • [26] P. Milbradt and T. Pick, Polytope finite elements, Internat. J. Numer. Methods Engrg. 73 (2008), no. 12, 1811-1835. MR 2397972 (2009a:65325), https://doi.org/10.1002/nme.2149
  • [27] A. Rand, A. Gillette, and C. Bajaj, Interpolation error estimates for mean value coordinates, Advances in Computational Mathematics 29 (2011), pp. 327-347.
  • [28] M. M. Rashid and M. Selimotic, A three-dimensional finite element method with arbitrary polyhedral elements, Internat. J. Numer. Methods Engrg. 67 (2006), no. 2, 226-252. MR 2241174 (2007a:65209), https://doi.org/10.1002/nme.1625
  • [29] Robin Sibson, A vector identity for the Dirichlet tessellation, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 1, 151-155. MR 549304 (81g:52008), https://doi.org/10.1017/S0305004100056589
  • [30] D. Sieger, P. Alliez, and M. Botsch, Optimizing voronoi diagrams for polygonal finite element computations, Proc. 19th Int. Meshing Roundtable (2010), 335-350.
  • [31] Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR 0443377 (56 #1747)
  • [32] N. Sukumar and E. A. Malsch, Recent advances in the construction of polygonal finite element interpolants, Arch. Comput. Methods Engrg. 13 (2006), no. 1, 129-163. MR 2283620 (2007m:65113), https://doi.org/10.1007/BF02905933
  • [33] N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2045-2066. MR 2101599 (2005h:65222), https://doi.org/10.1002/nme.1141
  • [34] A. Tabarraei and N. Sukumar, Application of polygonal finite elements in linear elasticity, Int. J. Comput. Methods 3 (2006), no. 4, 503-520. MR 2345859, https://doi.org/10.1142/S021987620600117X
  • [35] Rüdiger Verfürth, A note on polynomial approximation in Sobolev spaces, M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 715-719 (English, with English and French summaries). MR 1726481 (2000h:41016), https://doi.org/10.1051/m2an:1999159
  • [36] Eugene L. Wachspress, A rational finite element basis, Academic Press, Inc. [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1975. Mathematics in Science and Engineering, Vol. 114. MR 0426460 (54 #14403)
  • [37] M. Wicke, M. Botsch, and M. Gross, A finite element method on convex polyhedra, Comput. Graphics Forum 26 (2007), no. 3, 355-364.
  • [38] Jing Zhang and Fumio Kikuchi, Interpolation error estimates of a modified 8-node serendipity finite element, Numer. Math. 85 (2000), no. 3, 503-524. MR 1760933 (2001f:65141), https://doi.org/10.1007/s002110000104
  • [39] O. Zienkiewicz and R. Taylor, The finite element method, fifth ed., Butterworth-Heinemann, London, 2000.

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Additional Information

Alexander Rand
Affiliation: CD-adapco, 10800 Pecan Park Blvd, Austin, Texas 78750
Email: alexprand@gmaiil.com

Andrew Gillette
Affiliation: The University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
Email: agillette@math.arizona.edu

Chandrajit Bajaj
Affiliation: Department of Computer Science, The University of Texas at Austin, 2317 Speedway, Stop D9500, Austin, Texas 78712
Email: bajaj@cs.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02807-X
Keywords: Finite element, barycentric coordinates, serendipity
Received by editor(s): August 31, 2011
Received by editor(s) in revised form: July 19, 2012, and February 13, 2013
Published electronically: February 20, 2014
Additional Notes: This research was supported in part by NIH contracts R01-EB00487, R01-GM074258, and a grant from the UT-Portugal CoLab project.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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