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 $H^1$-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.