  We derive a Ewald decomposition for the Stokeslet in planar periodicity and a
novel PME-type O(N log N) method for the fast evaluation of the resulting sums.
The decomposition is the natural 2P counterpart to the classical 3P
decomposition by Hasimoto, and is given in an explicit form not found in the
literature. Truncation error estimates are provided to aid in selecting
parameters. The fast, PME-type, method appears to be the first fast method for
computing Stokeslet Ewald sums in planar periodicity, and has three attractive
properties: it is spectrally accurate; it uses the minimal amount of memory
that a gridded Ewald method can use; and provides clarity regarding numerical
errors and how to choose parameters. Analytical and numerical results are give
to support this. We explore the practicalities of the proposed method, and
survey the computational issues involved in applying it to 2-periodic boundary
integral Stokes problems.
