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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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:
  • 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:
  • MathSciNet review: 2772092