Conforming discrete Gradgrad-complexes in three dimensions
HTML articles powered by AMS MathViewer
- by Jun Hu and Yizhou Liang HTML | PDF
- Math. Comp. 90 (2021), 1637-1662 Request permission
Abstract:
In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one. These spaces can be used in the mixed form of the linearized Einstein-Bianchi system.References
- Arlen Anderson, Yvonne Choquet-Bruhat, and James W. York Jr., Einstein-Bianchi hyperbolic system for general relativity, Topol. Methods Nonlinear Anal. 10 (1997), no. 2, 353–373. Dedicated to Olga Ladyzhenskaya. MR 1634577, DOI 10.12775/TMNA.1997.037
- Helmut Friedrich, Hyperbolic reductions for Einstein’s equations, Classical Quantum Gravity 13 (1996), no. 6, 1451–1469. MR 1397127, DOI 10.1088/0264-9381/13/6/014
- Jun Hu, Finite element approximations of symmetric tensors on simplicial grids in $\Bbb R^n$: the higher order case, J. Comput. Math. 33 (2015), no. 3, 283–296. MR 3352360, DOI 10.4208/jcm.1412-m2014-0071
- Jun Hu and ShangYou Zhang, A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids, Sci. China Math. 58 (2015), no. 2, 297–307. MR 3301063, DOI 10.1007/s11425-014-4953-5
- Jun Hu and Shangyou Zhang, Finite element approximations of symmetric tensors on simplicial grids in $\Bbb {R}^n$: the lower order case, Math. Models Methods Appl. Sci. 26 (2016), no. 9, 1649–1669. MR 3529252, DOI 10.1142/S0218202516500408
- 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
- Vincent Quenneville-Belair, A New Approach to Finite Element Simulations of General Relativity, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–University of Minnesota. MR 3450102
- L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
- Alexander Ženíšek, Polynomial approximation on tetrahedrons in the finite element method, J. Approximation Theory 7 (1973), 334–351. MR 350260, DOI 10.1016/0021-9045(73)90036-1
- 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
Additional Information
- Jun Hu
- Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- MR Author ID: 714525
- Email: hujun@math.pku.edu.cn
- Yizhou Liang
- Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, Peoples’s Republic of China
- ORCID: 0000-0001-6318-1708
- Email: lyz2015@pku.edu.cn
- Received by editor(s): July 30, 2020
- Received by editor(s) in revised form: November 5, 2020
- Published electronically: March 24, 2021
- Additional Notes: The authors were supported by NSFC projects 11625101 and 11421101. The second author was supported by The Elite Program of Computational and Applied Mathematics for Ph.D. Candidates in Peking University.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1637-1662
- MSC (2020): Primary 65N30
- DOI: https://doi.org/10.1090/mcom/3628
- MathSciNet review: 4273111