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We present results for the decay constants of the $D$ and $D_s$ mesonscomputed in lattice QCD with $N_f=2+1$ dynamical flavours. The simulations arebased on RBC/UKQCD's domain wall ensembles with both physical and unphysicallight-quark masses and lattice spacings in the range 0.11--0.07$\,$fm. Weemploy 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 theCKM matrix elements.

The decay constants ${\mathbf{f_D}}$ and ${\mathbf{f_{D_{s}}}}$ in the continuum limit of ${\mathbf{N_f=2+1}}$ domain wall lattice QCD