  Domain decomposition methods are widely used for the numerical solution of
partial differential equations on high performance computers. We develop an
adjoint-based a posteriori error analysis for both multiplicative and additive
overlapping Schwarz domain decomposition methods. The numerical error in a
user-specified functional of the solution (quantity of interest) is decomposed
into contributions that arise as a result of the finite iteration between the
subdomains and from the spatial discretization. The spatial discretization
contribution is further decomposed into contributions arising from each
subdomain. This decomposition of the numerical error is used to construct a two
stage solution strategy that efficiently reduces the error in the quantity of
interest by adjusting the relative contributions to the error.
