Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence
Volume 271 / 2023
Studia Mathematica 271 (2023), 225-236
MSC: Primary 47A10; Secondary 47A58, 47B39.
DOI: 10.4064/sm221103-16-3
Published online: 8 May 2023
Abstract
We extend recent results on discrete approximations of the Laplacian in $\mathbb R^d$ 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.
