  We present results for the decay constants of the $D$ and $D_s$ mesons
computed in lattice QCD with $N_f=2+1$ dynamical flavours. The simulations are
based on RBC/UKQCD's domain wall ensembles with both physical and unphysical
light-quark masses and lattice spacings in the range 0.11--0.07$\,$fm. We
employ the domain wall discretisation for all valence quarks.
  The results in the continuum limit are
$f_D=208.7(2.8)_\mathrm{stat}\left(^{+2.1}_{-1.8}\right)_\mathrm{sys}\,\mathrm{MeV}$
and
$f_{D_{s}}=246.4(1.3)_\mathrm{stat}\left(^{+1.3}_{-1.9}\right)_\mathrm{sys}\,\mathrm{MeV}$
and
$f_{D_s}/f_D=1.1667(77)_\mathrm{stat}\left(^{+57}_{-43}\right)_\mathrm{sys}$.
Using these results in a Standard Model analysis we compute the predictions
$|V_{cd}|=0.2185(50)_\mathrm{exp}\left(^{+35}_{-37}\right)_\mathrm{lat}$ and
$|V_{cs}|=1.011(16)_\mathrm{exp}\left(^{+4}_{-9}\right)_\mathrm{lat}$ for the
CKM matrix elements.
