One-Sided p-Explicit Stability for High-Order Cut Finite Element Nitsche Methods
Abstract
We give a one-sided stability argument for high-order cut finite elements with symmetric Nitsche boundary conditions. On each boundary patch, an H1-projection splits the discrete function into a single polynomial and a stabilized residual. Inward segments control the polynomial inside the physical domain; complete-element estimates control only the residual. For smooth exterior boundaries and under a geometry-dependent mesh restriction, this gives an explicit normal-flux estimate with factor p2/h and a sufficient Nitsche penalty with the same scaling. The constants are independent of the cut position and polynomial degree. A neighboring-interval estimate quantifies the exponential cost of polynomial continuation and motivates avoiding it in the stability proof. We obtain hp-error estimates, with half a power of degree lost in energy and a full power in L2 relative to best approximation, and exponential degree convergence for analytic extensions on sufficiently fine meshes. Translated-disk experiments show bounded normalized flux constants and penalty thresholds over the tested degrees and cuts, and resolved convergence curves follow the predicted orders. Physical and extension-dependent stabilization errors are reported separately.
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BibTeX
@misc{burman2026onesided,
title = {{One-Sided $p$-Explicit Stability for High-Order Cut Finite Element Nitsche Methods}},
author = {Erik Burman and Peter Hansbo and Mats G. Larson and Karl Larsson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34853},
eprint = {2610.34853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34853},
}
RIS
TY - UNPB
TI - One-Sided p-Explicit Stability for High-Order Cut Finite Element Nitsche Methods
AU - Erik Burman
AU - Peter Hansbo
AU - Mats G. Larson
AU - Karl Larsson
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34853
UR - https://arxiv.org/abs/2610.34853
AB - We give a one-sided stability argument for high-order cut finite elements with symmetric Nitsche boundary conditions. On each boundary patch, an H^1-projection splits the discrete function into a single polynomial and a stabilized residual. Inward segments control the polynomial inside the physical domain; complete-element estimates control only the residual. For smooth exterior boundaries and under a geometry-dependent mesh restriction, this gives an explicit normal-flux estimate with factor p^2/h and a sufficient Nitsche penalty with the same scaling. The constants are independent of the cut position and polynomial degree. A neighboring-interval estimate quantifies the exponential cost of polynomial continuation and motivates avoiding it in the stability proof. We obtain hp-error estimates, with half a power of degree lost in energy and a full power in L^2 relative to best approximation, and exponential degree convergence for analytic extensions on sufficiently fine meshes. Translated-disk experiments show bounded normalized flux constants and penalty thresholds over the tested degrees and cuts, and resolved convergence curves follow the predicted orders. Physical and extension-dependent stabilization errors are reported separately.
ER -