{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fast-computation-of-high-frequency-dirichlet","title":"Fast computation of high frequency Dirichlet eigenmodes via the spectral flow of the interior Neumann-to-Dirichlet map","arxiv_id":"1112.5665","date":"2011-12-23","proceeding":null,"authors":["Alex H. Barnett","Andrew Hassell"],"abstract":"We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star-shaped domain in $\\mathbb{R}^d$, $d\\ge 2$. Conventional boundary-based methods require a root-search in eigenfrequency $k$, hence take $O(N^3)$ effort per eigenpair found, using dense linear algebra, where $N=O(k^{d-1})$ is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to $k$ the spectrum of a weighted interior Neumann-to-Dirichlet (NtD) operator for the Helmholtz equation. Approximations $\\hat{k}_j$ to the square-roots $k_j$ of all O(N) eigenvalues lying in $[k - \\epsilon, k]$, where $\\epsilon=O(1)$, are found with $O(N^3)$ effort. We prove an error estimate $$ |\\hat k_j - k_j| \\leq C \\Big(\\frac{\\epsilon^2}{k} + \\epsilon^3 \\Big), $$ with $C$ independent of $k$. We present a higher-order variant with eigenvalue error scaling empirically as $O(\\epsilon^5)$ and eigenfunction error as $O(\\epsilon^3)$, the former improving upon the 'scaling method' of Vergini--Saraceno. For planar domains ($d=2$), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For $d=2$ we compute robustly the spectrum of the NtD operator via potential theory, Nystr\\\"{o}m discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error $10^{-10}$, we show that the method is $10^3$ times faster than standard ones based upon a root-search.","url_abs":"https://arxiv.org/abs/1112.5665v1","url_pdf":"https://arxiv.org/pdf/1112.5665v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"fast-computation-of-high-frequency-dirichlet","repo_url":"https://github.com/ahbarnett/mpspack","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}