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. Note (Nov 27, 2024): A corrigendum has been added to the end of the PDF.

翻译：我们将近期关于$\mathbf{R}^d$中Laplace算子离散逼近的范数预解收敛结果，推广至半空间上的Dirichlet与Neumann Laplace算子情形。通过自然离散化与嵌入算子，将离散Dirichlet/Neumann Laplace算子的预解嵌入连续情形。证明了其与连续对应算子在网格尺寸上具有二阶收敛速率的范数预解收敛性。这些结果可有限速率地推广至具有实有界Hölder连续势的算子，以及Dirichlet/Neumann Laplace算子的某些函数（包括任意正实数幂）的情形。注（2024年11月27日）：PDF文档末尾已添加更正说明。