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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Divergence-free finite elements on tetrahedral grids for $k\ge 6$
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by Shangyou Zhang PDF
Math. Comp. 80 (2011), 669-695 Request permission

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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Additional 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