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Andrew Gillette
Serendipity and Tensor Product Affine Pyramid Finite Elements
SMAI-Journal of computational mathematics, 2 (2016), p. 215-228, doi: 10.5802/smai-jcm.14
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Class. Math.: 65N30, 41A20, 41A10
Keywords: Finite element methods; pyramid elements; rational functions

Abstract

Using the language of finite element exterior calculus, we define two families of $H^1$-conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements on the base while the second has matching polynomial traces with serendipity elements on the base. The second family is new to the literature and provides a robust approach for linking between Lagrange elements on tetrahedra and serendipity elements on affinely-mapped cubes while preserving continuity and approximation properties. We define shape functions and degrees of freedom for each family and prove unisolvence and polynomial reproduction results.

Bibliography

[1] M. Ainsworth, O. Davydov & L. Schumaker, “Bernstein–Bézier finite elements on tetrahedral–hexahedral–pyramidal partitions”, Computer Methods in Applied Mechanics and Engineering 304 (2016), p. 140-170 Article |  MR 3486313
[2] D.N. Arnold & G. Awanou, “The serendipity family of finite elements”, Foundations of Computational Mathematics 11 (2011) no. 3, p. 337-344 Article |  MR 2794906 |  Zbl 1218.65125
[3] D.N. Arnold, D. Boffi & F. Bonizzoni, “Finite element differential forms on curvilinear cubic meshes and their approximation properties”, Numerische Mathematik (2014), p. 1-20  MR 3296150
[4] D.N. Arnold, R. Falk & R. Winther, “Finite element exterior calculus, homological techniques, and applications”, Acta Numerica (2006), p. 1-155 Article |  MR 2269741 |  Zbl 1185.65204
[5] D.N. Arnold, R.S. Falk & R. Winther, “Finite element exterior calculus: from Hodge theory to numerical stability”, Bulletin of the American Mathematical Society 47 (2010) no. 2, p. 281-354 Article
[6] D.N. Arnold & A. Logg, “Periodic Table of the Finite Elements”, SIAM News 47 (2014. ) no. 9  MR 2594630 |  Zbl 1207.65134
[7] T.C. Baudouin, J.-F. Remacle, E. Marchandise, F. Henrotte & C. Geuzaine, “A frontal approach to hex-dominant mesh generation”, Advanced Modeling and Simulation in Engineering Sciences 1 (2014) no. 1, p. 1-30 Article
[8] G. Bedrosian, “Shape functions and integration formulas for three-dimensional finite element analysis”, International journal for numerical methods in engineering 35 (1992) no. 1, p. 95-108 Article |  Zbl 0772.65011
[9] L. Beirão Da Veiga, F. Brezzi, L.D. Marini & A. Russo, “Serendipity nodal VEM spaces”, Computers & Fluids in press (2016)
[10] M. Bergot, G. Cohen & M. Duruflé, “Higher-order finite elements for hybrid meshes using new nodal pyramidal elements”, Journal of Scientific Computing 42 (2010) no. 3, p. 345-381 Article |  MR 2585588 |  Zbl 1203.65243
[11] S. Brenner & L. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 2002  MR 1894376 |  Zbl 0804.65101
[12] J. Chan, Z. Wang, A. Modave & T. Remacle, “GPU-accelerated discontinuous Galerkin methods on hybrid meshes”, Journal of Computational Physics 318 (2016), p. 142-168 Article |  MR 3503992
[13] J. Chan & T. Warburton, “A comparison of high order interpolation nodes for the pyramid”, SIAM Journal on Scientific Computing 37 (2015) no. 5, p. A2151-A2170 Article |  MR 3392483
[14] J. Chan & T. Warburton, “hp-finite element trace inequalities for the pyramid”, Computers & Mathematics with Applications 69 (2015) no. 6, p. 510-517 Article |  MR 3315177
[15] J. Chan & T. Warburton, “A short note on a Bernstein–Bézier basis for the pyramid”, arXiv:1508.05609 (2015)  MR 3521548
[16] J. Chan & T. Warburton, “Orthogonal bases for vertex-mapped pyramids”, SIAM Journal on Scientific Computing 38 (2016) no. 2, p. A1146-A1170 Article |  MR 3488166
[17] S. Christiansen & A. Gillette, “Constructions of some minimal finite element systems”, ESAIM: Mathematical Modelling and Numerical Analysis 50 (2016) no. 3, p. 833-850 Article |  MR 3507275
[18] P. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics 40, SIAM, Philadelphia, PA, 2002  MR 1930132 |  Zbl 0999.65129
[19] J-L. Coulomb, F-X. Zgainski & Y. Maréchal, “A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements”, IEEE Trans. Magnetics 33 (1997) no. 2, p. 1362-1365 Article
[20] F. Fuentes, B. Keith, L. Demkowicz & S. Nagaraj, “Orientation embedded high order shape functions for the exact sequence elements of all shapes”, Computers and Mathematics with Applications 70 (2015) no. 4, p. 353 -458 Article |  MR 3372037
[21] T. Hughes, The finite element method, Prentice Hall Inc., Englewood Cliffs, NJ, 1987  MR 1008473 |  Zbl 0634.73056
[22] L. Liping, K.B. Davies, M. Krezek & L. Guan, “On higher order pyramidal finite elements”, Advances in Applied Mathematics and Mechanics 3 (2011) no. 2, p. 131-140 Article |  MR 2770081 |  Zbl 1262.65176
[23] L. Liu, K. B. Davies, K. Yuan & M. Křížek, “On symmetric pyramidal finite elements”, Dynamics of Continuous Discrete and Impulsive Systems Series B 11 (2004), p. 213-228  MR 2049777 |  Zbl 1041.65098
[24] J. Mandel, “Iterative solvers by substructuring for the $p$-version finite element method”, Computer Methods in Applied Mechanics and Engineering 80 (1990) no. 1-3, p. 117-128 Article |  MR 1067945 |  Zbl 0754.73086
[25] N. Nigam & J. Phillips, “High-order conforming finite elements on pyramids”, IMA Journal of Numerical Analysis 32 (2012) no. 2, p. 448-483 Article |  MR 2911396 |  Zbl 1241.65102
[26] N. Nigam & J. Phillips, “Numerical integration for high order pyramidal finite elements”, ESAIM: Mathematical Modelling and Numerical Analysis 46 (2012) no. 2, p. 239-263 Numdam |  MR 2855642 |  Zbl 1276.65083
[27] G. Strang & G. Fix, An analysis of the finite element method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973  MR 443377 |  Zbl 0356.65096
[28] B.A. Szabó & I. Babuška, Finite element analysis, Wiley-Interscience, 1991  Zbl 0792.73003
[29] F.D. Witherden & P.E. Vincent, “On the identification of symmetric quadrature rules for finite element methods”, Computers & Mathematics with Applications 69 (2015) no. 10, p. 1232-1241 Article |  MR 3333661
[30] F-X. Zgainski, J-L. Coulomb, Y. Maréchal, F. Claeyssen & X. Brunotte, “A new family of finite elements: the pyramidal elements”, IEEE Trans. Magnetics 32 (1996) no. 3, p. 1393-1396 Article