Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits

We derive low-order, inf–sup stable and divergence-free finite element approximations for the Stokes problem using Worsey–Farin splits in three dimensions and Powell–Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, whereas the pressure s...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2022-02, Vol.390, p.114444, Article 114444
Main Authors: Fabien, Maurice, Guzmán, Johnny, Neilan, Michael, Zytoon, Ahmed
Format: Article
Language:English
Subjects:
Apexes
Approximation
Divergence-free
Graph theory
Low-order
Mathematical analysis
Polynomials
Powell–Sabin
Worsey–Farin
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Summary:We derive low-order, inf–sup stable and divergence-free finite element approximations for the Stokes problem using Worsey–Farin splits in three dimensions and Powell–Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, whereas the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). We discuss implementation aspects that arise when coding the pressure space, and in particular, show that the pressure constraints can be enforced at an algebraic level.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2021.114444