A sharp Galerkin projection bound on nonobtuse triangles
Let $T \subset \mathbb{R}^2$ be a nondegenerate triangle whose interior angles are all at most $\pi/2$, and let $p \ge 0$ be an integer. Write $P_m(T)$ for the polynomials of total degree at most $m$ on $T$ and $\Pi_p$ for the componentwise $L^2(T)$-orthogonal projection onto $P_p(T)^2$. The Galerkin projection $G_{p+1} f \in P_{p+1}(T)$ of $f \in H^1(T)$ is defined by $$\int_T G_{p+1} f = \int_T f, \qquad \int_T \nabla G_{p+1} f \cdot \nabla q = \int_T \nabla f \cdot \nabla q \quad (q \in P_{p+1}(T)).$$ Carstensen, Gräßle and Tran showed that on every simplex the best constant $C$ in $\|\nabla(f - G_{p+1} f)\|_{L^2(T)} \le C \|(I - \Pi_p)\nabla f\|_{L^2(T)}$ for all $f \in H^1(T)$ is at least $1$, equals $1$ for $p = 0$, and, in dimensions two and three, is bounded independently of $p$. Their computations for several isosceles and right triangles indicate that $C^2$ tends to $2$ as $p \to \infty$, and that an interior angle larger than $\pi/2$ has a mild influence on its maximal value. The constant enters their guaranteed lower bounds for Laplace eigenvalues by a hybrid high-order method, and Carstensen, Gräßle and Pirch use the value $\sqrt{2}$ in benchmarks on right isosceles triangles.
Problem. Is $$\|\nabla(f - G_{p+1} f)\|_{L^2(T)} \le \sqrt{2}\, \|(I - \Pi_p)\nabla f\|_{L^2(T)}$$ for every integer $p \ge 0$, every such triangle $T$ and every $f \in H^1(T)$? No positive lower bound on the smallest angle of $T$ is assumed.
Source: Carstensen, Gräßle and Tran (2024), Conjecture 2.5.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Keywords
- AIM Open Applied Problems, Galerkin projections, hybrid high-order methods, nonobtuse triangles
References
@article{carstensen2024,
author = {Carstensen, Carsten and Gr{\"a}{\ss}le, Benedikt and Tran, Ngoc Tien},
title = {Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds},
journal = {Numer. Math.},
volume = {156},
number = {3},
year = {2024},
pages = {813--851},
doi = {10.1007/s00211-024-01407-w},
eprint = {2404.01228},
archivePrefix = {arXiv},
}
@article{carstensen2025,
author = {Carstensen, Carsten and Gr{\"a}{\ss}le, Benedikt and Pirch, Emilie},
title = {Comparison of guaranteed lower eigenvalue bounds with three skeletal schemes},
journal = {Comput. Methods Appl. Mech. Engrg.},
volume = {433},
year = {2025},
pages = {117477},
doi = {10.1016/j.cma.2024.117477},
}
@misc{aim-nonobtuse-galerkin-projection-constant,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A sharp Galerkin projection bound on nonobtuse triangles}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/569-nonobtuse-galerkin-projection-constant.md},
note = {Entry 569 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00364, "A sharp Galerkin projection bound on nonobtuse triangles", clopen.solutions.