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.
Funding source: NSF Office of the Director
Award Identifier / Grant number: 1522289
Funding source: NIH Office of the Director
Award Identifier / Grant number: R01-GM117594
Funding source: Sandia National Laboratories
Award Identifier / Grant number: BD-4485
Funding statement: AG was supported in part by NSF Award 1522289 and a J. Tinsley Oden Fellowship. CB was supported in part by a grant from NIH (R01-GM117594) and contract (BD-4485) from Sandia National Labs.
Acknowledgements
The authors would like to thank the anonymous referees for their helpful suggestions to improve the paper. Sean Stephens helped produce the figure.
References
[1] Abraham R., Marsden J. E. and Ratiu T., Manifolds, Tensor Analysis, and Applications, 2nd ed. Appl. Math. Sci. 75, Springer, New York, 1988. 10.1007/978-1-4612-1029-0Search in Google Scholar
[2] Alnæs M., Blechta J., Hake J., Johansson A., Kehlet B., Logg A., Richardson C., Ring J., Rognes M. E. and Wells G. N., The FEniCS Project version 1.5, Arch. Numer. Softw. 3 (2015), Paper No. 100. Search in Google Scholar
[3] Arnold D., Falk R. and Winther R., Finite element exterior calculus, homological techniques, and applications, Act. Numer. 15 (2006), 1–155. 10.1017/S0962492906210018Search in Google Scholar
[4] Arnold D., Falk R. and Winther R., Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 21–26, 1660–1672. 10.1016/j.cma.2008.12.017Search in Google Scholar
[5] Arnold D., Falk R. and Winther R., Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. 10.1090/S0273-0979-10-01278-4Search in Google Scholar
[6] Beirão da Veiga L., Brezzi F., Cangiani A., Manzini G., Marini L. D. and Russo A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199–214. 10.1142/S0218202512500492Search in Google Scholar
[7] Bossavit A., Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism, IEE Proc. A 135 (1988), no. 8, 493–500. 10.1049/ip-a-1.1988.0077Search in Google Scholar
[8] Bossavit A., A uniform rationale for Whitney forms on various supporting shapes, Math. Comput. Simulation 80 (2010), no. 8, 1567–1577. 10.1016/j.matcom.2008.11.005Search in Google Scholar
[9] Brezzi F., Douglas, Jr. J., Durán R. and Fortin M., Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. 10.1007/BF01396752Search in Google Scholar
[10] Brezzi F., Douglas, Jr. J. and Marini L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. 10.1007/BF01389710Search in Google Scholar
[11] Chen W. and Wang Y., Minimal degree H(curl) and H(div) conforming finite elements on polytopal meshes, preprint 2015, http://arxiv.org/abs/1502.01553. 10.1090/mcom/3152Search in Google Scholar
[12] Christiansen S. H., A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 739–757. 10.1142/S021820250800284XSearch in Google Scholar
[13] Christiansen S. H. and Winther R., Smoothed projections into finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. 10.1090/S0025-5718-07-02081-9Search in Google Scholar
[14] Clément P., Approximation by finite element functions using local regularization, Rev. Franc. Automat. Inform. Rech. Operat. 9 (1975), no. R-2, 77–84. 10.1051/m2an/197509R200771Search in Google Scholar
[15] Ern A. and Guermond J.-L., Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer, New York, 2004. 10.1007/978-1-4757-4355-5Search in Google Scholar
[16] Euler T., Schuhmann R. and Weiland T., Polygonal finite elements, IEEE Trans. Magn. 42 (2006), no. 4, 675–678. 10.1109/TMAG.2006.871375Search in Google Scholar
[17] Farin G., Surfaces over Dirichlet tessellations, Comput. Aided Geom. Design 7 (1990), no. 1–4, 281–292. 10.1016/0167-8396(90)90036-QSearch in Google Scholar
[18] Floater M., Mean value coordinates, Comput. Aided Geom. Design 20 (2003), no. 1, 19–27. 10.1016/S0167-8396(03)00002-5Search in Google Scholar
[19] Floater M., Gillette A. and Sukumar N., Gradient bounds for Wachspress coordinates on polytopes, SIAM J. Numer. Anal. 52 (2014), no. 1, 515–532. 10.1137/130925712Search in Google Scholar
[20] Floater M., Hormann K. and Kós G., A general construction of barycentric coordinates over convex polygons, Adv. Comput. Math. 24 (2006), no. 1, 311–331. 10.1007/s10444-004-7611-6Search in Google Scholar
[21] Floater M., Kós G. and Reimers M., Mean value coordinates in 3D, Comput. Aided Geom. Design 22 (2005), no. 7, 623–631. 10.1016/j.cagd.2005.06.004Search in Google Scholar
[22] Gillette A. and Bajaj C., A generalization for stable mixed finite elements, Proceedings of the 14th ACM Symposium on Solid and Physical Modeling (SPM ’10), ACM, New York (2010), 41–50. 10.1145/1839778.1839785Search in Google Scholar
[23] Gillette A. and Bajaj C., Dual formulations of mixed finite element methods with applications, Comput. Aided Des. 43 (2011), no. 10, 1213–1221. 10.1016/j.cad.2011.06.017Search in Google Scholar PubMed PubMed Central
[24] Gillette A., Rand A. and Bajaj C., Error estimates for generalized barycentric coordinates, Adv. Comput. Math. 37 (2012), no. 3, 417–439. 10.1007/s10444-011-9218-zSearch in Google Scholar PubMed PubMed Central
[25] Gradinaru V., Whitney elements on sparse grids, Ph.D. thesis, Universität Tübingen, Tübingen, 2002. Search in Google Scholar
[26] Gradinaru V. and Hiptmair R., Whitney elements on pyramids, Electron. Trans. Numer. Anal. 8 (1999), 154–168. Search in Google Scholar
[27] Hirani A. N., Discrete exterior calculus, Dissertation, California Institute of Technology, Pasedena, 2003. Search in Google Scholar
[28] Hormann K. and Sukumar N., Maximum entropy coordinates for arbitrary polytopes, Comp. Graph. Forum 27 (2008), no. 5, 1513–1520. 10.1111/j.1467-8659.2008.01292.xSearch in Google Scholar
[29] Joshi P., Meyer M., DeRose T., Green B. and Sanocki T., Harmonic coordinates for character articulation, ACM Trans. Graph. 26 (2007), Article ID 71. 10.1145/1275808.1276466Search in Google Scholar
[30] Ju T., Schaefer S., Warren J. D. and Desbrun M., A geometric construction of coordinates for convex polyhedra using polar duals, Proceedings of the Third Eurographics Symposium on Geometry Processing (SGP ’05), Eurographics Association, Aire-la-Ville (2015), 181–186. Search in Google Scholar
[31] Klausen R., Rasmussen A. and Stephansen A., Velocity interpolation and streamline tracing on irregular geometries, Comput. Geosci. 16 (2011), no. 2, 1–16. 10.1007/s10596-011-9256-0Search in Google Scholar
[32] Lipnikov K., Manzini G. and Shashkov M., Mimetic finite difference method, J. Comput. Phys. 257B (2014), 1163–1227. 10.1016/j.jcp.2013.07.031Search in Google Scholar
[33] Manson J. and Schaefer S., Moving least squares coordinates, Comp. Graph. Forum 29 (2010), no. 5, 1517–1524. 10.1111/j.1467-8659.2010.01760.xSearch in Google Scholar
[34] Manzini G., Russo A. and Sukumar N., New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1665–1699. 10.1142/S0218202514400065Search in Google Scholar
[35] Martin S., Kaufmann P., Botsch M., Wicke M. and Gross M., Polyhedral finite elements using harmonic basis functions, Comp. Graph. Forum 27 (2008), no. 5, 1521–1529. 10.1111/j.1467-8659.2008.01293.xSearch in Google Scholar
[36] Milbradt P. and Pick T., Polytope finite elements, Internat. J. Numer. Methods Engrg. 73 (2008), no. 12, 1811–1835. 10.1002/nme.2149Search in Google Scholar
[37]
Nédélec J.-C.,
Mixed finite elements in
[38]
Nédélec J.-C.,
A new family of mixed finite elements in
[39] Rand A., Average interpolation under the maximum angle condition, SIAM J. Numer. Anal. 50 (2012), no. 5, 2538–2559. 10.1137/10081842XSearch in Google Scholar
[40] Rand A., Gillette A. and Bajaj C., Interpolation error estimates for mean value coordinates, Adv. Comput. Math. 39 (2013), 327–347. 10.1007/s10444-012-9282-zSearch in Google Scholar PubMed PubMed Central
[41] Rand A., Gillette A. and Bajaj C., Quadratic serendipity finite elements on polygons using generalized barycentric coordinates, Math. Comp. 83 (2014), no. 290, 2691–2716. 10.1090/S0025-5718-2014-02807-XSearch in Google Scholar
[42] Rashid M. and Selimotic M., A three-dimensional finite element method with arbitrary polyhedral elements, Internat. J. Numer. Methods Engrg. 67 (2006), no. 2, 226–252. 10.1002/nme.1625Search in Google Scholar
[43] Raviart P.-A. and Thomas J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods (Rome 1975), Lecture Notes in Math. 606, Springer, Berlin (1977), 292–315. 10.1007/BFb0064470Search in Google Scholar
[44] Scott L. and Zhang S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. 10.1090/S0025-5718-1990-1011446-7Search in Google Scholar
[45] Sibson R., A vector identity for the Dirichlet tessellation, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 1, 151–155. 10.1017/S0305004100056589Search in Google Scholar
[46] Sukumar N., Construction of polygonal interpolants: A maximum entropy approach, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2159–2181. 10.1002/nme.1193Search in Google Scholar
[47] Sukumar N. and Malsch E. A., Recent advances in the construction of polygonal finite element interpolants, Arch. Comput. Methods Eng. 13 (2006), no. 1, 129–163. 10.1007/BF02905933Search in Google Scholar
[48] Sukumar N. and Tabarraei A., Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2045–2066. 10.1002/nme.1141Search in Google Scholar
[49] Wachspress E., A Rational Finite Element Basis, Math. Sci. Eng. 114, Academic Press, New York, 1975. Search in Google Scholar
[50] Wachspress E., Barycentric coordinates for polytopes, Comput. Math. Appl. 61 (2011), no. 11, 3319–3321. 10.1016/j.camwa.2011.04.032Search in Google Scholar
[51] Warren J., Barycentric coordinates for convex polytopes, Adv. Comput. Math. 6 (1996), no. 1, 97–108. 10.1007/BF02127699Search in Google Scholar
[52] Warren J., Schaefer S., Hirani A. N. and Desbrun M., Barycentric coordinates for convex sets, Adv. Comput. Math. 27 (2007), no. 3, 319–338. 10.1007/s10444-005-9008-6Search in Google Scholar
[53] Whitney H., Geometric Integration Theory, Princeton University Press, Princeton, 1957. 10.1515/9781400877577Search in Google Scholar
[54] Wicke M., Botsch M. and Gross M., A finite element method on convex polyhedra, Comp. Graph. Forum 26 (2007), no. 3, 355–364. 10.1111/j.1467-8659.2007.01058.xSearch in Google Scholar
© 2016 by De Gruyter
