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

Horia Cornean, Henrik Garde, Arne Jensen

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 Hölder 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.

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

Abstract

We extend recent results on discrete approximations of the Laplacian in 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. Note (Nov 27, 2024): A corrigendum has been added to the end of the PDF.
Paper Structure (8 sections, 9 theorems, 66 equations)

This paper contains 8 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extension and restriction operators
  4. Embedding and discretization operators
  5. Laplacians
  6. Results
  7. Adding a potential
  8. Functions of Dirichlet and Neumann Laplacians

Key Result

Theorem 1

Let $K\subset\mathbf{C}\setminus[0,\infty)$ be compact. Then there exists $C>0$ such that and for $0<h\leq1$ and $z\in K$.

Theorems & Definitions (17)

  • Theorem
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 3.2
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • ...and 7 more