Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods
Abstract
We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, for any window of N consecutive levels, the sum of squared Galerkin increments is bounded by the squared error at the beginning of the window times Nσ, where σ<1. The exponent σ and the prefactor can be chosen explicitly in terms of the uniform projection bound. The proof uses Hilbert-space projection estimates, dyadic iteration, and duality, without relying on matrix factorizations. As a corollary, we obtain explicit growth bounds for finite block-LU factors. In the symmetric indefinite case, the positive and negative spectral splittings of the Galerkin detail spaces yield a characterization of full quasi-orthogonality in terms of uniform boundedness of the associated finite-window sign operators. We also construct a self-adjoint involution and a nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. For every 0<α<1, the construction yields a hierarchy whose finite-window constants have growth of exact order Nα, along with an explicit target whose Galerkin increments are not square summable.
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BibTeX
@misc{gantumur2026projection,
title = {{Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods}},
author = {Tsogtgerel Gantumur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16265},
eprint = {2609.16265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16265},
}
RIS
TY - UNPB
TI - Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods
AU - Tsogtgerel Gantumur
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.16265
UR - https://arxiv.org/abs/2609.16265
AB - We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, for any window of N consecutive levels, the sum of squared Galerkin increments is bounded by the squared error at the beginning of the window times N^σ, where σ<1. The exponent σ and the prefactor can be chosen explicitly in terms of the uniform projection bound. The proof uses Hilbert-space projection estimates, dyadic iteration, and duality, without relying on matrix factorizations. As a corollary, we obtain explicit growth bounds for finite block-LU factors. In the symmetric indefinite case, the positive and negative spectral splittings of the Galerkin detail spaces yield a characterization of full quasi-orthogonality in terms of uniform boundedness of the associated finite-window sign operators. We also construct a self-adjoint involution and a nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. For every 0<α<1, the construction yields a hierarchy whose finite-window constants have growth of exact order N^α, along with an explicit target whose Galerkin increments are not square summable.
ER -