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

TL;DR

This paper proves that discrete approximations of Dirichlet and Neumann Laplacians on a half-space converge in norm resolvent sense to their continuous versions, with a quadratic rate, and extends these results to include potentials and functions of the operators.

Contribution

It extends recent discrete approximation results to half-space Laplacians, establishing quadratic convergence rates and including potentials and operator functions.

Findings

Norm resolvent convergence with quadratic rate

Extension to operators with bounded H"older continuous potentials

Inclusion of functions of the Laplacians, including positive powers

Abstract

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.