  The DCA$^+$ algortihm was recently introduced to extend the dynamic cluster
approximation (DCA) with a continuous lattice self-energy in order to achieve
better convergence with cluster size. Here we extend the DCA$^+$ algorithm to
the calculation of two-particle correlation functions by introducing
irreducible vertex functions with continuous momentum dependence consistent
with the DCA$^+$ self-energy. This enables a significantly more controlled and
reliable study of phase transitions than with the DCA. We test the new method
by calculating the superconducting transition temperature $T_{c}$ in the
attractive Hubbard model and show that it reproduces previous high-precision
determinantal quantum Monte Carlo results. We then calculate $T_c$ in the doped
repulsive Hubbard model, for which previous DCA calculations could only access
the weak-coupling ($U=4t$) regime for large clusters. We show that the new
algorithm provides access to much larger clusters and delivers asymptotically
converged results for $T_c$ for both the weak ($U=4t$) and intermediate
($U=7t$) coupling regimes, and thereby enables the accurate determination of
the exact infinite cluster size result.
