  We solve first-kind Fredholm boundary integral equations arising from
Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions
with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov
method takes as discretization elements pushed-forward weighted azimuthal
projections of standard spherical harmonics onto the canonical disk. We show
that these bases allow for spectral convergence and provide fully discrete
error analysis. Numerical experiments support our claims, with results
comparable to Nystr\"om-type and $hp$-methods.
