Hi all.
I am looking for clarifications on the hexahedral mixed displacement-pressure elements.
1.) What exactly is the Taylor-Hood hexahedral element? Is it hex element with 20 nodes for displacement and 8 nodes for pressure or 27 nodes for displacement and 8 nodes for pressure? (Considering continuous interpolation for pressure.)
2.) Are both 20/8 or 27/8 elements inf-sup stable? If not, which one is inf-sup stable and which is not? Is there any good reference on this topic?
Thanks.
I would recommend Section 4.5.2.3 of the following book:
@book{reddy2010finite, title={The finite element method in heat transfer and fluid dynamics}, author={Reddy, Junuthula Narasimha and Gartling, David K}, year={2010}, publisher={CRC press}}
Here, it is said that both Q_2 Q_1 elements with 20 or 27 nodes for the velocities (for fluid dynamics) are LBB-stable.
Taylor-Hood elements often refer to [P_k]^d P_{k-1} or [Q_k]^d Q_{k-1} elements, where k is the polynomial order and d is the dimension.
But there are also other element types that are LBB-stable, e.g., [P_1-bubble]^2 P_1. See for example:
@article{suli2013brief, title={A brief excursion into the mathematical theory of mixed finite element methods}, author={S{\"u}li, ENDRE}, journal={Lecture Notes, University of Oxford}, pages={24--29}, year={2013}}
Hope that helps!
Hi Andreas Hessenthaler . Thank you for your response. I appreciate it very much.
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