Finite element de Rham and Stokes complexes in three dimensions

Chen, Long; Huang, Xuehai

doi:10.1090/mcom/3859

Finite element de Rham and Stokes complexes in three dimensions
HTML articles powered by AMS MathViewer

by Long Chen and Xuehai Huang

Math. Comp. 93 (2024), 55-110

DOI: https://doi.org/10.1090/mcom/3859

Published electronically: June 7, 2023

HTML | PDF | Request permission

Abstract:

Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. $H(\operatorname {div})$-conforming finite elements and $H(\operatorname {curl})$-conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the $H(\operatorname {div})$-conforming finite elements is established, and the exactness of these finite element complexes is proven.

References

J. Argyris, I. Fried, and D. Scharpf, The TUBA family of plate elements for the matrix displacement method, Aero. J. Roy. Aero. Soc. 72 (1968), 701–709.
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
Douglas N. Arnold, Finite element exterior calculus, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 93, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018. MR 3908678, DOI 10.1137/1.9781611975543.ch1
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 21-26, 1660–1672. MR 2517938, DOI 10.1016/j.cma.2008.12.017
Douglas N. Arnold and Kaibo Hu, Complexes from complexes, Found. Comput. Math. 21 (2021), no. 6, 1739–1774. MR 4343022, DOI 10.1007/s10208-021-09498-9
Francesca Bonizzoni and Guido Kanschat, $H^1$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes, Calcolo 58 (2021), no. 2, Paper No. 18, 29. MR 4244253, DOI 10.1007/s10092-021-00409-6
F. Bonizzoni and G. Kanschat, A tensor-product finite element cochain complex with arbitrary continuity, Preprint, arXiv:2207.00309, 2022.
James H. Bramble and Miloš Zlámal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 282540, DOI 10.1090/S0025-5718-1970-0282540-0
Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
A. Buffa, C. de Falco, and G. Sangalli, IsoGeometric Analysis: stable elements for the 2D Stokes equation, Internat. J. Numer. Methods Fluids 65 (2011), no. 11-12, 1407–1422. MR 2808112, DOI 10.1002/fld.2337
L. Chen and X. Huang, Complexes from complexes: finite element complexes in three dimensions, arXiv preprint arXiv:2211.08656, 2022.
Long Chen and Xuehai Huang, Decoupling of mixed methods based on generalized Helmholtz decompositions, SIAM J. Numer. Anal. 56 (2018), no. 5, 2796–2825. MR 3852718, DOI 10.1137/17M1145872
L. Chen and X. Huang, Finite elements for divdiv-conforming symmetric tensors, arXiv preprint arXiv:2005.01271, 2020.
L. Chen and X. Huang, Geometric decomposition of div-conforming finite element tensors, part I: Vector and matrix functions, arXiv preprint arXiv:2112.14351, 2021.
L. Chen and X. Huang, Finite element complexes in two dimensions, arXiv preprint arXiv:2206.00851, 2022.
Long Chen and Xuehai Huang, A finite element elasticity complex in three dimensions, Math. Comp. 91 (2022), no. 337, 2095–2127. MR 4451457, DOI 10.1090/mcom/3739
Long Chen and Xuehai Huang, Finite elements for $\textrm {div\,div}$ conforming symmetric tensors in three dimensions, Math. Comp. 91 (2022), no. 335, 1107–1142. MR 4405490, DOI 10.1090/mcom/3700
S. H. Christiansen, J. Gopalakrishnan, J. Guzmán, and K. Hu, A discrete elasticity complex on three-dimensional Alfeld splits, arXiv preprint arXiv:2009.07744, 2020.
Snorre H. Christiansen, Jun Hu, and Kaibo Hu, Nodal finite element de Rham complexes, Numer. Math. 139 (2018), no. 2, 411–446. MR 3802677, DOI 10.1007/s00211-017-0939-x
Snorre H. Christiansen and Kaibo Hu, Generalized finite element systems for smooth differential forms and Stokes’ problem, Numer. Math. 140 (2018), no. 2, 327–371. MR 3851060, DOI 10.1007/s00211-018-0970-6
Charles K. Chui and Ming Jun Lai, Multivariate vertex splines and finite elements, J. Approx. Theory 60 (1990), no. 3, 245–343. MR 1042653, DOI 10.1016/0021-9045(90)90063-V
Martin Costabel and Alan McIntosh, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265 (2010), no. 2, 297–320. MR 2609313, DOI 10.1007/s00209-009-0517-8
John A. Evans and Thomas J. R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations, Math. Models Methods Appl. Sci. 23 (2013), no. 8, 1421–1478. MR 3048532, DOI 10.1142/S0218202513500139
Richard S. Falk and Michael Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal. 51 (2013), no. 2, 1308–1326. MR 3045658, DOI 10.1137/120888132
Guosheng Fu, Johnny Guzmán, and Michael Neilan, Exact smooth piecewise polynomial sequences on Alfeld splits, Math. Comp. 89 (2020), no. 323, 1059–1091. MR 4063312, DOI 10.1090/mcom/3520
Johnny Guzmán and Michael Neilan, A family of nonconforming elements for the Brinkman problem, IMA J. Numer. Anal. 32 (2012), no. 4, 1484–1508. MR 2991835, DOI 10.1093/imanum/drr040
Jun Hu and Yizhou Liang, Conforming discrete Gradgrad-complexes in three dimensions, Math. Comp. 90 (2021), no. 330, 1637–1662. MR 4273111, DOI 10.1090/mcom/3628
Jun Hu, Yizhou Liang, and Rui Ma, Conforming finite element divdiv complexes and the application for the linearized Einstein-Bianchi system, SIAM J. Numer. Anal. 60 (2022), no. 3, 1307–1330. MR 4431967, DOI 10.1137/21M1404235
J. Hu, Y. Liang, R. Ma, and M. Zhang, New conforming finite element divdiv complexes in three dimensions, arXiv preprint arXiv:2204.07895, 2022.
J. Hu, T. Lin, and Q. Wu, A construction of ${C}^r$ conforming finite element spaces in any dimension, arXiv:2103.14924, 2021.
Jun Hu, Rui Ma, and Min Zhang, A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids, Sci. China Math. 64 (2021), no. 12, 2793–2816. MR 4343068, DOI 10.1007/s11425-020-1883-9
Kaibo Hu, Qian Zhang, and Zhimin Zhang, A family of finite element Stokes complexes in three dimensions, SIAM J. Numer. Anal. 60 (2022), no. 1, 222–243. MR 4369574, DOI 10.1137/20M1358700
X. Huang, Nonconforming finite element Stokes complexes in three dimensions, Sci. China Math. (2023), https://doi.org/10.1007/s11425-021-2026-7.
X. Huang and C. Zhang, Robust mixed finite element methods for a quad-curl singular perturbation problem, arXiv preprint arXiv:2206.10864, 2022.
Ming-Jun Lai and Larry L. Schumaker, Spline functions on triangulations, Encyclopedia of Mathematics and its Applications, vol. 110, Cambridge University Press, Cambridge, 2007. MR 2355272, DOI 10.1017/CBO9780511721588
Ming-Jun Lai and Larry L. Schumaker, Trivariate $C^r$ polynomial macroelements, Constr. Approx. 26 (2007), no. 1, 11–28. MR 2310685, DOI 10.1007/s00365-006-0631-x
P. Le Tallec, A mixed finite element approximation of the Navier-Stokes equations, Numer. Math. 35 (1980), no. 4, 381–404. MR 593835, DOI 10.1007/BF01399007
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
Michael Neilan, Discrete and conforming smooth de Rham complexes in three dimensions, Math. Comp. 84 (2015), no. 295, 2059–2081. MR 3356019, DOI 10.1090/S0025-5718-2015-02958-5
Michael Neilan, The Stokes complex: a review of exactly divergence-free finite element pairs for incompressible flows, 75 years of mathematics of computation, Contemp. Math., vol. 754, Amer. Math. Soc., [Providence], RI, [2020] ©2020, pp. 141–158. MR 4132120, DOI 10.1090/conm/754/15142
Michael Neilan and Duygu Sap, Stokes elements on cubic meshes yielding divergence-free approximations, Calcolo 53 (2016), no. 3, 263–283. MR 3541155, DOI 10.1007/s10092-015-0148-x
Michael Neilan and Duygu Sap, Macro Stokes elements on quadrilaterals, Int. J. Numer. Anal. Model. 15 (2018), no. 4-5, 729–745. MR 3789588
R. A. Nicolaides, On a class of finite elements generated by Lagrange interpolation. II, SIAM J. Numer. Anal. 10 (1973), 182–189. MR 317512, DOI 10.1137/0710019
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin-New York, 1977, pp. 292–315. MR 483555
L. R. Scott and M. Vogelius, Conforming finite element methods for incompressible and nearly incompressible continua, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983) Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 221–244. MR 818790, DOI 10.1051/m2an/1985190101111
Rolf Stenberg, A nonstandard mixed finite element family, Numer. Math. 115 (2010), no. 1, 131–139. MR 2594344, DOI 10.1007/s00211-009-0272-0
Xue-Cheng Tai and Ragnar Winther, A discrete de Rham complex with enhanced smoothness, Calcolo 43 (2006), no. 4, 287–306. MR 2283095, DOI 10.1007/s10092-006-0124-6
Alexander Ženíšek, Interpolation polynomials on the triangle, Numer. Math. 15 (1970), 283–296. MR 275014, DOI 10.1007/BF02165119
A. Ženíšek, Tetrahedral finite $C^{(m)}$-elements, Acta Univ. Carolin. Math. Phys. 15 (1974), no. 1-2, 189–193. MR 395160, DOI 10.1016/0016-0032(64)90077-8
Q. Zhang and Z. Zhang, A family of curl-curl conforming finite elements on tetrahedral meshes, CSIAM Trans. Appl. Math. 1 (2020), no. 4, 639–663.
Shangyou Zhang, A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math. 59 (2009), no. 1, 219–233. MR 2474112, DOI 10.1016/j.apnum.2008.02.002
Shangyou Zhang, Divergence-free finite elements on tetrahedral grids for $k\geq 6$, Math. Comp. 80 (2011), no. 274, 669–695. MR 2772092, DOI 10.1090/S0025-5718-2010-02412-3
Shang You Zhang, A family of differentiable finite elements on simplicial grids in four space dimensions, Math. Numer. Sin. 38 (2016), no. 3, 309–324 (Chinese, with English and Chinese summaries). MR 3617203
Bin Zheng, Qiya Hu, and Jinchao Xu, A nonconforming finite element method for fourth order curl equations in $\Bbb {R}^{3}$, Math. Comp. 80 (2011), no. 276, 1871–1886. MR 2813342, DOI 10.1090/S0025-5718-2011-02480-4