  An efficient, accurate, and flexible numerical method is proposed for the
solution of the swimming problem of one or more autophoretic particles in the
purely-diffusive limit. The method relies on successive boundary element
solutions of the Laplacian and the Stokes flow equations using regularised
Green's functions for swift, simple implementations, an extension of the
well-known method of "regularised stokeslets" for Stokes flow problems. The
boundary element method is particularly suitable for phoretic problems, since
no quantities in the domain bulk are required to compute the swimming velocity.
For time-dependent problems, the method requires no re-meshing and simple
boundaries such as a plane wall may be added at no increase to the size of the
linear system through the method of images. The method is validated against two
classical examples for which an analytical or semi-analytical solution is
known, a two-sphere system and a Janus particle, and provides a rigorous
computational pipeline to address further problems with complex geometry and
multiple bodies.

Fluid Dynamics;Soft Condensed Matter