Axisymmetric Stokes equations in polygonal domains: regularity and finite element approximations

We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain $Ω$. 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 $\maK^m_{μ, 1}(Ω)$. 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.