  We present the first lattice QCD study of coupled isoscalar
$\pi\pi,K\overline{K},\eta\eta$ $S$- and $D$-wave scattering extracted from
discrete finite-volume spectra computed on lattices which have a value of the
quark mass corresponding to $m_\pi\sim391$ MeV. In the $J^P=0^+$ sector we find
analogues of the experimental $\sigma$ and $f_0(980)$ states, where the
$\sigma$ appears as a stable bound-state below $\pi\pi$ threshold, and, similar
to what is seen in experiment, the $f_0(980)$ manifests itself as a dip in the
$\pi\pi$ cross section in the vicinity of the $K\overline{K}$ threshold. For
$J^P=2^+$ we find two states resembling the $f_2(1270)$ and $f_2'(1525)$,
observed as narrow peaks, with the lighter state dominantly decaying to
$\pi\pi$ and the heavier state to $K\overline{K}$. The presence of all these
states is determined rigorously by finding the pole singularity content of
scattering amplitudes, and their couplings to decay channels are established
using the residues of the poles.
