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Conforming and divergence-free Stokes elements on general triangular meshes


Authors: Johnny Guzmán and Michael Neilan
Journal: Math. Comp. 83 (2014), 15-36
MSC (2010): Primary 76M10, 65N30, 65N12
DOI: https://doi.org/10.1090/S0025-5718-2013-02753-6
Published electronically: July 25, 2013
MathSciNet review: 3120580
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Abstract: We present a family of conforming finite elements for the Stokes problem on general triangular meshes in two dimensions. The lowest order case consists of enriched piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure. We show that the elements satisfy the inf-sup condition and converges with order $ k$ for both the velocity and pressure. Moreover, the pressure space is exactly the divergence of the corresponding space for the velocity. Therefore the discretely divergence-free functions are divergence-free pointwise. We also show how the proposed elements are related to a class of $ C^1$ elements through the use of a discrete de Rham complex.


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Additional Information

Johnny Guzmán
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: johnnyguzman@brown.edu

Michael Neilan
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: neilan@pitt.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02753-6
Keywords: Finite elements, Stokes, conforming, divergence-free
Received by editor(s): October 3, 2011
Received by editor(s) in revised form: March 29, 2012
Published electronically: July 25, 2013
Additional Notes: The first author was supported by the National Science Foundation through grant number DMS-0914596
The second author was supported by the National Science Foundation through grant number DMS-1115421
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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