On Stability, Accuracy, and Fast Solvers for Finite Element Approximations of the Axisymmetric Stokes Problem by Hood-Taylor Elements

We provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric Stokes problem by continuous piecewise polynomials of degree κ+ 1 for the velocity and continuous piecewise polynomials of degree κ for the pressure with any κ ≥ 1. New techniques are designed so that in this perspective, by a simple transformation, the existing theory developed in three dimensional Cartesian coordinates can be effectively exploited. In fact, this perspective provides a new way of developing theories for the axisymmetric Stokes problems and it can be applied potentially to other problems as well. A simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems. Sample numerical experiments have been presented as well to confirm the theoretical results.