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The high-frequency Helmholtz equation on the entire space is truncated into abounded domain using the perfectly matched layer (PML) technique andsubsequently, discretized by the higher-order finite element method (FEM) andthe continuous interior penalty finite element method (CIP-FEM). By formulatingan elliptic problem involving a linear combination of a finite number ofeigenfunctions related to the PML differential operator, a wave-number-explicitdecomposition lemma is proved for the PML problem, which implies that the PMLsolution can be decomposed into a non-oscillating elliptic part and anoscillating but analytic part. The preasymptotic error estimates in the energynorm for both the $p$-th order CIP-FEM and FEM are proved to be $C_1(kh)^p +C_2k(kh)^{2p} +C_3 E^{\rm PML}$ under the mesh condition that $k^{2p+1}h^{2p}$is sufficiently small, where $k$ is the wave number, $h$ is the mesh size, and$E^{\rm PML}$ is the PML truncation error which is exponentially small. Inparticular, the dependences of coefficients $C_j~(j=1,2)$ on the source $f$ areimproved. Numerical experiments are presented to validate the theoreticalfindings, illustrating that the higher-order CIP-FEM can greatly reduce thepollution errors.

Higher-order FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation