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Spectral Convergence of the Discrete Steklov Problem
Abstract
We study the eigenvalues and eigenvectors of the Schur complement S of the graph Laplacian, defined on vertices sampled from a domain with C2 boundary. We use a thin boundary layer to approximate the continuum boundary as the number of vertices increases and rely on transportation theory to justify the consistency of a discrete Steklov problem. In particular, S acts on vectors defined over the shrinking boundary layer but behaves asymptotically like a known operator. In this paper, we use a variational argument to determine conditions for which the eigenvalues and eigenvectors of S converge to the eigenvalues and eigenfunctions of a Dirichlet-to-Neumann map, and we provide evidence of this spectral convergence through a numerical example.
Related
- Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator
- Regularized solutions for an inverse problem governed by a parabolic Steklov equation
- The exterior Steklov problem for Euclidean domains
- Equivariant Steklov homogenisation and stability of spectral representations
- Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains
BibTeX
@misc{lyons2026spectral,
title = {{Spectral Convergence of the Discrete Steklov Problem}},
author = {Andrew Lyons},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40920},
eprint = {2610.40920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40920},
}
RIS
TY - UNPB
TI - Spectral Convergence of the Discrete Steklov Problem
AU - Andrew Lyons
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40920
UR - https://arxiv.org/abs/2610.40920
AB - We study the eigenvalues and eigenvectors of the Schur complement S of the graph Laplacian, defined on vertices sampled from a domain with C^2 boundary. We use a thin boundary layer to approximate the continuum boundary as the number of vertices increases and rely on transportation theory to justify the consistency of a discrete Steklov problem. In particular, S acts on vectors defined over the shrinking boundary layer but behaves asymptotically like a known operator. In this paper, we use a variational argument to determine conditions for which the eigenvalues and eigenvectors of S converge to the eigenvalues and eigenfunctions of a Dirichlet-to-Neumann map, and we provide evidence of this spectral convergence through a numerical example.
ER -