Abstract
We introduce new Hermite style and Bernstein style geometric decompositions of the cubic serendipity finite element spaces \({\mathcal S}_3(I^2)\) and \({\mathcal S}_3(I^3)\), as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337–344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces—12 instead of 16 for the square and 32 instead of 64 for the cube—yet are still guaranteed to obtain cubic order a priori error estimates in \(H^1\) norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.
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References
Arnold, D., Awanou, G.: The serendipity family of finite elements. Found. Comput. Math. 11(3), 337–344 (2011)
Arnold, D.N., Awanou, G.: Finite element differential forms on cubical meshes. Math. Comput. 83, 1551–1570 (2014)
Arnold, D., Falk, R., Winther, R.: Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198(21–26), 1660–1672 (2009)
Bangerth, W., Hartmann, R., Kanschat, G.: Deal. ii—A general-purpose object-oriented finite element library. ACM Trans. Math. Softw. (TOMS) 33(4), 24–es (2007)
Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40, 2nd edn. SIAM, Philadelphia (2002)
Ciarlet, P., Raviart, P.: General Lagrange and Hermite interpolation in \(\mathbb{R}^n\) with applications to finite element methods. Arch. Ration. Mechan. Anal. 46(3), 177–199 (1972)
Cottrell, J., Hughes, T., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)
Evans, J.A., Hughes, T.J.R.: Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Numer. Math. 123(2), 259–290 (2013)
Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. AK Peters, Wellesley (1993)
Hughes, T.: The Finite Element Method. Prentice Hall, Englewood Cliffs (1987)
Mandel, J.: Iterative solvers by substructuring for the \(p\)-version finite element method. Comput. Methods Appl. Mech. Eng. 80(1–3), 117–128 (1990)
Mortenson, M.: Geometric Modeling, 3rd edn. Wiley, New York (2006)
Rand, A., Gillette, A., Bajaj, C.: Quadratic serendipity finite elements on polygons using generalized barycentric coordinates. Math. Comput. http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2014-02807-X/home.html (2014)
Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)
Szabó, B., Babuška, I.: Finite Element Analysis. Wiley-Interscience, New York (1991)
Zhang, Y., Liang, X., Ma, J., Jing, Y., Gonzales, M.J., Villongco, C., Krishnamurthy, A., Frank, L.R., Nigam, V., Stark, P., Narayan, S.M., McCulloch, A.D.: An atlas-based geometry pipeline for cardiac hermite model construction and diffusion tensor reorientation. Med. Image Anal. 16(6), 1130–1141 (2012)
Acknowledgments
Support for this work was provided in part by NSF Award 0715146 and the National Biomedical Computation Resource while the author was at the University of California, San Diego.
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Gillette, A. (2014). Hermite and Bernstein Style Basis Functions for Cubic Serendipity Spaces on Squares and Cubes. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_7
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DOI: https://doi.org/10.1007/978-3-319-06404-8_7
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