A First-Order Exactly Incompressible Finite Element for Axisymmetric Fluid Flow

We discuss a finite element for incompressible flow of a fluid in axisymmetric geometry. Let u denote a velocity field; define the "reduced velocity" vuvia vu(r, z) = r u(r, z). The element we address is a composite quadrilateral element obtained by dividing each quadrilateral into four tr...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1996-10, Vol.33 (5), p.1736-1758
Main Authors: Bernstein, Barry, Feigl, Kathleen A., Olsen, Elwood T.
Format: Article
Language:English
Subjects:
Approximation
Computational methods in fluid dynamics
Computational techniques
Exact sciences and technology
Finite-element and galerkin methods
Fluid dynamics
Fundamental areas of phenomenology (including applications)
Geometry
Incompressibility
Mathematical methods in physics
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Physics
Radial velocity
Rectangles
Sciences and techniques of general use
Symmetry
Triangles
Triangulation
Velocity
Velocity distribution
Vertices
Viscoelasticity
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Summary:We discuss a finite element for incompressible flow of a fluid in axisymmetric geometry. Let u denote a velocity field; define the "reduced velocity" vuvia vu(r, z) = r u(r, z). The element we address is a composite quadrilateral element obtained by dividing each quadrilateral into four triangles by drawing diagonals. The reduced velocity vuis approximated by a piecewise linear function which is linear on each triangle, and the pressure p is approximated by a step function which is constant on each triangle. The velocities u are therefore approximated by piecewise rational functions rather than piecewise polynomials. The resulting approximation is shown to be conforming, and weak incompressibility is shown to imply pointwise incompressibility for the element. Rigorous, though possibly suboptimal, convergence results for the element are obtained.
ISSN:0036-1429
1095-7170
DOI:10.1137/S0036142994243095