Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence

We extend recent results on discrete approximations of the Laplacian in Rd 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 Holder continuous potential, as well as certain functions of the Dirichlet/Neumann Laplacians, including any positive real power.