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/UKQCDs 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.
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