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

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  • Horia Cornean, Aalborg University, Denmark
  • Henrik Garde
  • Arne Jensen, Aalborg University, Denmark
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 Hölder continuous potential, as well as certain functions of the Dirichlet/Neumann Laplacians, including any positive real power.
Original languageEnglish
JournalStudia Mathematica
Pages (from-to)225-236
Number of pages12
Publication statusPublished - 2023

    Research areas

  • norm resolvent convergence, Dirichlet Laplacian, Neumann Laplacian, lattice

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