  We study the regularity and finite element approximation of the axisymmetric
Stokes problem on a polygonal domain $\Omega$. In particular, taking into
account the singular coefficients in the equation and non-smoothness of the
domain, we establish the well-posedness and full regularity of the solution in
new weighted Sobolev spaces $\maK^m_{\mu, 1}(\Omega)$. Using our a priori
results, we give a specific construction of graded meshes on which the
Taylor-Hood mixed method approximates singular solutions at the optimal
convergence rate. Numerical tests are presented to confirm the theoretical
results in the paper.
