We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of $S^3$; 2) in
$H^3$, the smaller volume is at least 85% that of the larger. A balancing argument and asymptotic analysis reduce the problem in $S^3$ and $H^3$ to some computer checking. The computer analysis has been designed and fully implemented for both spaces.
