A new div-div-conforming symmetric tensor finite element space with applications to the biharmonic equation

Chen, Long; Huang, Xuehai

doi:10.1090/mcom/3957

A new div-div-conforming symmetric tensor finite element space with applications to the biharmonic equation
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by Long Chen and Xuehai Huang;

Math. Comp. 94 (2025), 33-72

DOI: https://doi.org/10.1090/mcom/3957

Published electronically: March 20, 2024

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Abstract:

A new $H(\operatorname {div}\operatorname {div})$-conforming finite element is presented, which avoids the need for supersmoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and $C^0$ discontinuous Galerkin methods for the biharmonic equation are derived.

References

D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
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
F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys. 131 (1997), no. 2, 267–279. MR 1433934, DOI 10.1006/jcph.1996.5572
H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci. 2 (1980), no. 4, 556–581. MR 595625, DOI 10.1002/mma.1670020416
Susanne C. Brenner and Li-Yeng Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput. 22/23 (2005), 83–118. MR 2142191, DOI 10.1007/s10915-004-4135-7
F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations 16 (2000), no. 4, 365–378. MR 1765651, DOI 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y
Long Chen, Jun Hu, and Xuehai Huang, Multigrid methods for Hellan-Herrmann-Johnson mixed method of Kirchhoff plate bending problems, J. Sci. Comput. 76 (2018), no. 2, 673–696. MR 3817803, DOI 10.1007/s10915-017-0636-z
L. Chen and X. Huang, Finite elements for divdiv-conforming symmetric tensors, Preprint, arXiv:2005.01271, 2020.
Long Chen and Xuehai Huang, Nonconforming virtual element method for $2m$th order partial differential equations in $\Bbb {R}^n$, Math. Comp. 89 (2020), no. 324, 1711–1744. MR 4081916, DOI 10.1090/mcom/3498
L. Chen and X. Huang, $H(\textrm {div})$-conforming finite element tensors, Preprint, arXiv:2112.14351, 2021.
L. Chen and X. Huang, Complexes from complexes: Finite element complexes in three dimensions, Preprint, arXiv:2211.08656, 2022.
Long Chen and Xuehai Huang, Finite elements for div- and divdiv-conforming symmetric tensors in arbitrary dimension, SIAM J. Numer. Anal. 60 (2022), no. 4, 1932–1961. MR 4458899, DOI 10.1137/21M1433708
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
Snorre H. Christiansen and Kaibo Hu, Finite element systems for vector bundles: elasticity and curvature, Found. Comput. Math. 23 (2023), no. 2, 545–596. MR 4568199, DOI 10.1007/s10208-022-09555-x
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Bernardo Cockburn and Jayadeep Gopalakrishnan, Incompressible finite elements via hybridization. II. The Stokes system in three space dimensions, SIAM J. Numer. Anal. 43 (2005), no. 4, 1651–1672. MR 2182143, DOI 10.1137/040610659
M. I. Comodi, The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing, Math. Comp. 52 (1989), no. 185, 17–29. MR 946601, DOI 10.1090/S0025-5718-1989-0946601-7
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 34, 3669–3750. MR 1915664, DOI 10.1016/S0045-7825(02)00286-4
B. Fraeijs de Veubeke, Displacement and equilibrium models in the finite element method, Stress analysis (O. C. Zienkiewicz and G. Holister, eds.), John Wiley & Sons, 1965.
T. Führer and N. Heuer, Mixed finite elements for Kirchhoff-Love plate bending, Preprint, arXiv:2305.08693, 2023.
Thomas Führer, Norbert Heuer, and Antti H. Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, Math. Comp. 88 (2019), no. 318, 1587–1619. MR 3925478, DOI 10.1090/mcom/3381
K. Hellan, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method, Acta Polytechnica Scandinavica. Civil Engineering and Building Construction Series, vol. 46, Norges Tekniske Vitenskapsakademi, Trondheim, 1967.
L. R. Herrmann, Finite element bending analysis for plates, J. Eng. Mech. Division 93 (1967), no. EM5, 49–83.
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, Yizhou Liang, R. Ma, and M. Zhang, New conforming finite element divdiv complexes in three dimensions, Preprint, arXiv:2204.07895, 2022.
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
Xuehai Huang and Jianguo Huang, A reduced local $C^0$ discontinuous Galerkin method for Kirchhoff plates, Numer. Methods Partial Differential Equations 30 (2014), no. 6, 1902–1930. MR 3267358, DOI 10.1002/num.21883
Claes Johnson, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math. 21 (1973), 43–62. MR 388807, DOI 10.1007/BF01436186
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
Dirk Pauly and Walter Zulehner, The divDiv-complex and applications to biharmonic equations, Appl. Anal. 99 (2020), no. 9, 1579–1630. MR 4113080, DOI 10.1080/00036811.2018.1542685
Astrid Pechstein and Joachim Schöberl, Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity, Math. Models Methods Appl. Sci. 21 (2011), no. 8, 1761–1782. MR 2826472, DOI 10.1142/S0218202511005568
Astrid S. Pechstein and Joachim Schöberl, An analysis of the TDNNS method using natural norms, Numer. Math. 139 (2018), no. 1, 93–120. MR 3785664, DOI 10.1007/s00211-017-0933-3
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
Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, DOI 10.1051/m2an/1991250101511
Ming Wang and Jinchao Xu, The Morley element for fourth order elliptic equations in any dimensions, Numer. Math. 103 (2006), no. 1, 155–169. MR 2207619, DOI 10.1007/s00211-005-0662-x
Garth N. Wells and Nguyen Tien Dung, A $C^0$ discontinuous Galerkin formulation for Kirchhoff plates, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 35-36, 3370–3380. MR 2335272, DOI 10.1016/j.cma.2007.03.008
Xiu Ye and Shangyou Zhang, A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes, SIAM J. Numer. Anal. 58 (2020), no. 5, 2572–2588. MR 4150276, DOI 10.1137/19M1276601
Peng Zhu, Shenglan Xie, and Xiaoshen Wang, A stabilizer-free $C^0$ weak Galerkin method for the biharmonic equations, Sci. China Math. 66 (2023), no. 3, 627–646. MR 4549959, DOI 10.1007/s11425-021-1947-0