## Finite elements for div div conforming symmetric tensors in three dimensions

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Long Chen and Xuehai Huang
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## Abstract:

Finite element spaces on a tetrahedron are constructed for $\operatorname {div}\operatorname {div}$ -conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the $\operatorname {div}\operatorname {div}$ Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensor spaces are derived from the polynomial complexes. Second, traces for the $\operatorname {div}\operatorname {div}$ operator are characterized through a Green’s identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Besides, a finite element for $symcurl$-conforming trace-free tensors is constructed following the same approach. Putting all together, a finite element $\operatorname {div}\operatorname {div}$ complex, as well as the bubble functions complex, in three dimensions is established.## References

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## Additional Information

**Long Chen**- Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
- MR Author ID: 735779
- ORCID: 0000-0002-7345-5116
- Email: chenlong@math.uci.edu
**Xuehai Huang**- Affiliation: School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China
- MR Author ID: 854280
- ORCID: 0000-0003-2966-7426
- Email: huang.xuehai@sufe.edu.cn
- Received by editor(s): August 16, 2020
- Received by editor(s) in revised form: February 19, 2021, March 3, 2021, July 19, 2021, and September 9, 2021
- Published electronically: December 10, 2021
- Additional Notes: The first author was supported by NSF DMS-2012465, and in part by DMS-1913080. The second author was supported by the National Natural Science Foundation of China Projects 11771338 and 12171300, the Natural Science Foundation of Shanghai 21ZR1480500 and the Fundamental Research Funds for the Central Universities 2019110066

The second author is the corresponding author - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 1107-1142 - MSC (2020): Primary 65N30, 65N12, 65N22
- DOI: https://doi.org/10.1090/mcom/3700
- MathSciNet review: 4405490