## Divergence-free finite elements on tetrahedral grids for $k\ge 6$

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- by Shangyou Zhang;
- Math. Comp.
**80**(2011), 669-695 - DOI: https://doi.org/10.1090/S0025-5718-2010-02412-3
- Published electronically: August 26, 2010
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## Abstract:

It was shown two decades ago that the $P_k$-$P_{k-1}$ mixed element on triangular grids, approximating the velocity by the continuous $P_k$ piecewise polynomials and the pressure by the discontinuous $P_{k-1}$ piecewise polynomials, is stable for all $k\ge 4$, provided the grids are free of a nearly-singular vertex. The problem with the method in 3D was posted then and remains open. The problem is solved partially in this work. It is shown that the $P_k$-$P_{k-1}$ element is stable and of optimal order in approximation, on a family of uniform tetrahedral grids, for all $k\ge 6$. The analysis is to be generalized to non-uniform grids, when we can deal with the complicity of 3D geometry.

For the divergence-free elements, the finite element spaces for the pressure can be avoided in computation, if a classic iterated penalty method is applied. The finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity. Numerical tests are provided.

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## Bibliographic Information

**Shangyou Zhang**- Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
- MR Author ID: 261174
- Email: szhang@udel.edu
- Received by editor(s): June 18, 2008
- Received by editor(s) in revised form: January 25, 2010
- Published electronically: August 26, 2010
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**80**(2011), 669-695 - MSC (2010): Primary 65N30, 76M10, 76D07
- DOI: https://doi.org/10.1090/S0025-5718-2010-02412-3
- MathSciNet review: 2772092