  Elastic scattering amplitudes for $I=0$ $DK$ and $I=0,1$ $D\bar{K}$ are
computed in $S$, $P$ and $D$ partial waves using lattice QCD with light-quark
masses corresponding to $m_\pi = 239$ MeV and $m_\pi = 391$ MeV. The $S$-waves
contain interesting features including a near-threshold $J^P=0^+$ bound state
in $I=0$ $DK$, corresponding to the $D_{s0}^\ast(2317)$, with an effect that is
clearly visible above threshold, and suggestions of a $0^+$ virtual bound state
in $I=0$ $D\bar{K}$. The $S$-wave $I=1$ $D\bar{K}$ amplitude is found to be
weakly repulsive. The computed finite-volume spectra also contain a
deeply-bound $D^\ast$ vector resonance, but negligibly small $P$-wave $DK$
interactions are observed in the energy region considered; the $P$ and $D$-wave
$D\bar{K}$ amplitudes are also small. There is some evidence of $1^+$ and $2^+$
resonances in $I=0$ $DK$ at higher energies.
