A FIRST-ORDER EXACTLY INCOMPRESSIBLE FINITE-ELEMENT FOR AXISYMMETRICAL FLUID-FLOW

We discuss a finite element for incompressible flow of a fluid in axis ymmetric 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 quadrilate ral 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 approxim ated by piecewise rational functions rather than piecewise polynomials . The resulting approximation is shown to be conforming, and weak inco mpressibility is shown to imply pointwise incompressibility for the el ement. Rigorous, though possibly suboptimal, convergence results for t he element are obtained.