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

11/03/2022

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by   Horia Cornean, et al.

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