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.

翻译：我们将在$\mathbf{R}^d$上具有范数豫解收敛性的拉普拉斯算子离散近似的最新结果推广至半空间上Dirichlet和Neumann拉普拉斯算子的相应情形。利用自然的离散化和嵌入算子，将离散Dirichlet/Neumann拉普拉斯算子的豫解嵌入到连续空间中。证明了其以网格尺寸的二次速率范数豫解收敛于连续情形。这些结果以有限速率进一步推广至包含实值、有界且Hölder连续势的算子，以及Dirichlet/Neumann拉普拉斯算子的某些函数（包括任意正实数幂）。