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

Author: Shangyou Zhang
Journal: Math. Comp. 80 (2011), 669-695
MSC (2010): Primary 65N30, 76M10, 76D07
Published electronically: August 26, 2010
MathSciNet review: 2772092
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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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Shangyou Zhang
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716

Keywords: Mixed finite elements, Stokes equations, divergence-free element, tetrahedral grids.
Received by editor(s): June 18, 2008
Received by editor(s) in revised form: January 25, 2010
Published electronically: August 26, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.