  Einstein's system of equations in the ADM decomposition involves two
subsystems of equations: evolution equations and constraint equations. For
numerical relativity, one typically solves the constraint equations only on the
initial time slice, and then uses the evolution equations to advance the
solution in time. Our interest is in the case when the spatial domain is
bounded and appropriate boundary conditions are imposed. A key difficulty,
which we address in this thesis, is what boundary conditions to place at the
artificial boundary that lead to long time stable numerical solutions. We
develop an effective technique for finding well-posed constraint preserving
boundary conditions for constrained first order symmetric hyperbolic systems.
By using this technique, we study the preservation of constraints by some first
order symmetric hyperbolic formulations of Einstein's equations derived from
the ADM decomposition linearized around Minkowski spacetime with arbitrary
lapse and shift perturbations, and the closely related question of their
equivalence with the linearized ADM system. Our main result is the finding of
well-posed maximal nonnegative constraint preserving boundary conditions for
each of the first order symmetric hyperbolic formulations under investigation,
for which the unique solution of the corresponding initial boundary value
problem provides a solution to the linearized ADM system on polyhedral domains.
