  An extension of the Method of Regularized Stokeslets (MRS) in three
dimensions is developed for triangulated surfaces with a piecewise linear force
distribution. The method extends the regularized Stokeslet segment methodology
used for piecewise linear curves. By using analytic integration of the
regularized Stokeslet kernel over the triangles, the regularization parameter
$\epsilon$ is effectively decoupled from the spatial discretization of the
surface. This is in contrast to the usual implementation of the method in which
the regularization parameter is chosen for accuracy reasons to be about the
same size as the spatial discretization. The validity of the method is
demonstrated through several examples, including the flow around a rigidly
translating/rotating sphere, a rotating spheroid, and the squirmer model for
self-propulsion. Notably, second order convergence in the spatial
discretization for fixed $\epsilon$ is demonstrated. Considerations of mesh
design and choice of regularization parameter are discussed, and the
performance of the method is compared with existing quadrature-based
implementations.
