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The dynamical mean-field theory (DMFT) combined with the fluctuation exchange(FLEX) method, namely FLEX+DMFT, is an approach for correlated electron systemsto incorporate both local and non-local long-range correlations in aself-consistent manner. We formulate FLEX+DMFT in a systematic way startingfrom a Luttinger-Ward functional, and apply it to study the $d$-wavesuperconductivity in the two-dimensional repulsive Hubbard model. The criticaltemperature ($T_c$) curve obtained in the FLEX+DMFT exhibits a dome structureas a function of the filling, which has not been clearly observed in the FLEXapproach alone. We trace back the origin of the dome to the local vertexcorrection from DMFT that renders a filling dependence in the FLEX self-energy.We compare the results with those of GW+DMFT, where the $T_c$-dome structure isqualitatively reproduced due to the same vertex correction effect, but acrucial difference from FLEX+DMFT is that $T_c$ is always estimated below theN\'{e}el temperature in GW+DMFT. The single-particle spectral function obtainedwith FLEX+DMFT exhibits a double-peak structure as a precursor of the Hubbardbands at temperature above $T_c$.

FLEX+DMFT approach to the $d$-wave superconducting phase diagram of the two-dimensional Hubbard model