Continuous meets discrete: geometry dependence of the performance of iterative solutions for nonsymmetric PDEs in terms of Laplace eigenvalues
Abstract
It is a classical fact in spectral geometry that the eigenvalues of the Dirichlet Laplacian depend continuously on the geometry of the underlying domain, under suitable geometric restrictions. We show that this theoretical property is reflected in a concrete numerical situation: the convergence rate of iterative solvers for convection-diffusion problems remains stable under deformations of the domain. The connection is a superlinear estimate for preconditioned GMRES/GCR in which the relative residual errors are bounded by an explicit expression of the eigenvalues of the Laplacian, or of a symmetric positive elliptic operator in divergence form. We prove that this bound depends continuously on the domain, uniformly in the iteration index, under either regularity hypotheses on the boundary or a topological constraint in dimension two. We also establish corresponding statements on the discrete level. Two examples delimit the theory: domains bounded by a fractal curve, where continuity still holds, and perforated domains, where it fails. The theoretical results, including both examples with rough domains, are confirmed by numerical experiments.
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BibTeX
@misc{francesconi2026continuous,
title = {{Continuous meets discrete: geometry dependence of the performance of iterative solutions for nonsymmetric PDEs in terms of Laplace eigenvalues}},
author = {F. Francesconi and J. Karátson and B. Lados and J. Rowlett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14946},
eprint = {2610.14946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14946},
}
RIS
TY - UNPB
TI - Continuous meets discrete: geometry dependence of the performance of iterative solutions for nonsymmetric PDEs in terms of Laplace eigenvalues
AU - F. Francesconi
AU - J. Karátson
AU - B. Lados
AU - J. Rowlett
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14946
UR - https://arxiv.org/abs/2610.14946
AB - It is a classical fact in spectral geometry that the eigenvalues of the Dirichlet Laplacian depend continuously on the geometry of the underlying domain, under suitable geometric restrictions. We show that this theoretical property is reflected in a concrete numerical situation: the convergence rate of iterative solvers for convection-diffusion problems remains stable under deformations of the domain. The connection is a superlinear estimate for preconditioned GMRES/GCR in which the relative residual errors are bounded by an explicit expression of the eigenvalues of the Laplacian, or of a symmetric positive elliptic operator in divergence form. We prove that this bound depends continuously on the domain, uniformly in the iteration index, under either regularity hypotheses on the boundary or a topological constraint in dimension two. We also establish corresponding statements on the discrete level. Two examples delimit the theory: domains bounded by a fractal curve, where continuity still holds, and perforated domains, where it fails. The theoretical results, including both examples with rough domains, are confirmed by numerical experiments.
ER -