  As predicted by quantum chromodynamics(QCD), around $T\sim 190$ MeV in the
early universe, the QCD transition occurs during which the quarks are combined
into the massive hadrons. This process reduces the effective relativistic
degree of freedom, and causes a change in the expansion behavior of the
universe. Similarly, the $e^+e^-$ annihilation occurred around $T\sim 0.5$ Mev
has the same kind of effect. Besides, the dark energy also drives the present
stage accelerating expansion. We study these combined effects on the relic
gravitational waves (RGWs). In our treatment, the QCD transition and the
$e^+e^-$ annihilation, each is respectively represented by a short period of
expansion inserted into the radiation era. Incorporating these effects, the
equation of RGWs is analytically solved for a spatially flat universe, evolving
from the inflation up to the current acceleration, and the spectrum of RGWs is
obtained, covering the whole range of frequency $>10^{-19}$ Hz. It is found
that the QCD transition causes a reduction of the amplitude of RGWs by $\sim
20%$ in the range $>10^{-9} $ Hz, and the $e^+e^-$ annihilation causes a
reduction $\sim 10%$ in the range $>10^{-12} $ Hz. In the presence of the dark
energy, the combination of the QCD transition and the $e^+e^-$ annihilation,
causes a larger reduction of the amplitude by $\sim 30% $ for the range
$>10^{-9} $ Hz, which covers the bands of operation of LIGO and LISA. By
analysis, it is shown that RGWs will be difficult to detect by the present
LIGO, but can be tested by LISA for certain inflationary models.
