  We extend recent results on discrete approximations of the Laplacian in
$\mathbf{R}^d$ with norm resolvent convergence to the corresponding results for
Dirichlet and Neumann Laplacians on a half-space. The resolvents of the
discrete Dirichlet/Neumann Laplacians are embedded into the continuum using
natural discretization and embedding operators. Norm resolvent convergence to
their continuous counterparts is proven with a quadratic rate in the mesh size.
These results generalize with a limited rate to also include operators with a
real, bounded, and H\"older continuous potential, as well as certain functions
of the Dirichlet/Neumann Laplacians, including any positive real power.
