  We present scalable implementations of spectral-element-based Schwarz
overlapping (overset) methods for the incompressible Navier-Stokes (NS)
equations. Our SEM-based overset grid method is implemented at the level of the
NS equations, which are advanced independently within separate subdomains using
interdomain velocity and pressure boundary-data exchanges at each timestep or
sub-timestep. Central to this implementation is a general, robust, and scalable
interpolation routine, {\em gslib-findpts}, that rapidly determines the
computational coordinates (processor $p$, element number $e$, and local
coordinates $(r,s,t) \in \hat{\Omega} := [-1,1]^3$) for any arbitrary point
$\mathbf{x}^* =(x^*,y^*,z^*) \in \Omega \subset {\rm I\!R}^3$. The
communication kernels in $gslib$ execute with at most $\log P$ complexity for
$P$ MPI ranks, have scaled to $P > 10^6$, and obviate the need for development
of any additional MPI-based code for the Schwarz implementation. The original
interpolation routine has been extended to account for multiple overlapping
domains. The new implementation discriminates the possessing subdomain by
distance to the domain boundary, such that the interface boundary data is taken
from the inner-most interior points. We present application of this approach to
several heat transfer and fluid dynamic problems, discuss the
computation/communication complexity and accuracy of the approach, and present
performance measurements for $P > 12,000$.
