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Nearly sharp W1,∞ stability of polynomial spectral Ritz projections
Abstract
We establish nearly sharp W1,∞ stability estimates for the Legendre spectral Ritz projection associated with Laplacian on a square domain. Its norm grows proportionally to the square root of the polynomial degree, up to a cubic logarithmic factor. The upper bound uses an integral representation of the discrete gradient projection in terms of one-dimensional semigroup kernels and a rank-two Duhamel comparison between the mass-matrix semigroup and its infinite-dimensional counterpart. The lower bound follows from the endpoint concentration of the Legendre kernel. As an application, we derive gradient maximum-norm error estimates for the Poisson and heat equations, with spatial factors optimal up to logarithmic terms.
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BibTeX
@misc{guo2026nearly,
title = {{Nearly sharp $W^1,\infty$ stability of polynomial spectral Ritz projections}},
author = {Yongxing Guo and Yuwen Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19531},
eprint = {2610.19531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19531},
}
RIS
TY - UNPB
TI - Nearly sharp W^1,∞ stability of polynomial spectral Ritz projections
AU - Yongxing Guo
AU - Yuwen Li
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19531
UR - https://arxiv.org/abs/2610.19531
AB - We establish nearly sharp W^1,∞ stability estimates for the Legendre spectral Ritz projection associated with Laplacian on a square domain. Its norm grows proportionally to the square root of the polynomial degree, up to a cubic logarithmic factor. The upper bound uses an integral representation of the discrete gradient projection in terms of one-dimensional semigroup kernels and a rank-two Duhamel comparison between the mass-matrix semigroup and its infinite-dimensional counterpart. The lower bound follows from the endpoint concentration of the Legendre kernel. As an application, we derive gradient maximum-norm error estimates for the Poisson and heat equations, with spatial factors optimal up to logarithmic terms.
ER -