  We present a Schur complement Domain Decomposition (DD) algorithm for the
solution of frequency domain multiple scattering problems. Just as in the
classical DD methods we (1) enclose the ensemble of scatterers in a domain
bounded by an artificial boundary, (2) we subdivide this domain into a
collection of nonoverlapping subdomains so that the boundaries of the
subdomains do not intersect any of the scatterers, and (3) we connect the
solutions of the subproblems via Robin boundary conditions matching on the
common interfaces between subdomains. We use subdomain Robin-to-Robin maps to
recast the DD problem as a sparse linear system whose unknown consists of Robin
data on the interfaces between subdomains---two unknowns per interface. The
Robin-to-Robin maps are computed in terms of well-conditioned boundary integral
operators. Unlike classical DD, we do not reformulate the Domain Decomposition
problem in the form a fixed point iteration, but rather we solve the ensuing
linear system by Gaussian elimination of the unknowns corresponding to inner
interfaces between subdomains via Schur complements. Once all the unknowns
corresponding to inner subdomains interfaces have been eliminated, we solve a
much smaller linear system involving unknowns on the inner and outer artificial
boundary. We present numerical evidence that our Schur complement DD algorithm
can produce accurate solutions of very large multiple scattering problems that
are out of reach for other existing approaches.

Numerical Analysis