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'' v(u) via v(u)(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 v, is 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.