Wavenumber-Explicit Quasi-Optimal Error Analysis of Linear CIP-FEM for the Helmholtz Equation on Nonconvex Polygonal Obstacle Domains

We study the linear continuous interior penalty finite element method (CIP-FEM) for the two-dimensional Helmholtz equation on nonconvex polygonal obstacle domains with mixed Dirichlet and impedance boundary conditions. Re-entrant corners reduce the global regularity below H2(Ω) and require a corner-sensitive treatment of the CIP stabilization term. Using a wavenumber-explicit regular–singular decomposition, we establish exact consistency of the CIP formulation under this reduced regularity. We also prove the critical-order penalty-seminorm estimate |IhSj|J ≲ hαj for the Scott–Zhang quasi-interpolant of the cut-off corner singular functions. Combining this estimate with endpoint Scott–Zhang approximation and wavenumber-explicit bounds for the regular and singular components yields infvh ∈ Vh∥u−vh∥h,k ≲ (kh + kα−1/2hα)∥f∥0,Ω. A Schatz-type duality argument then yields the corresponding quasi-optimal error estimate ∥u−uh∥h,k ≲ (kh + kα−1/2hα)∥f∥0,Ω under the sufficient resolution condition k2h ≤ δ, where δ is independent of k and h. Discrete uniqueness is proved independently of this condition. Numerical experiments support the predicted corner-singularity behavior; for the tested problems with the fixed penalty value γ = 0.1, they show smaller errors than the standard FEM on some relatively coarse meshes in the high-wavenumber pre-asymptotic regime. The present analysis does not separately identify an additive pollution error term.