The high-frequency Helmholtz equation on the entire space is truncated into a bounded domain using the perfectly matched layer (PML) technique and subsequently, discretized by the higher-order finite element method (FEM) and the continuous interior penalty finite element method (CIP-FEM). By formulating an elliptic problem involving a linear combination of a finite number of eigenfunctions related to the PML differential operator, a wave-number-explicit decomposition lemma is proved for the PML problem, which implies that the PML solution can be decomposed into a non-oscillating elliptic part and an oscillating but analytic part. The preasymptotic error estimates in the energy norm 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. In particular, the dependences of coefficients $C_j~(j=1,2)$ on the source $f$ are improved. Numerical experiments are presented to validate the theoretical findings, illustrating that the higher-order CIP-FEM can greatly reduce the pollution errors.

翻译：全空间上的高频亥姆霍兹方程通过完美匹配层（PML）技术截断为有界区域，随后采用高阶有限元法（FEM）和连续内罚有限元法（CIP-FEM）进行离散。通过构造一个涉及PML微分算子的有限个特征函数线性组合的椭圆问题，证明了PML问题的波数显式分解引理，该引理表明PML解可分解为非振荡椭圆部分和振荡解析部分。在网格条件$k^{2p+1}h^{2p}$足够小的前提下，证明了$p$阶CIP-FEM和FEM在能量范数下的预渐近误差估计为$C_1(kh)^p + C_2k(kh)^{2p} +C_3 E^{\rm PML}$，其中$k$为波数，$h$为网格尺寸，$E^{\rm PML}$为指数级小的PML截断误差。特别地，系数$C_j~(j=1,2)$对源项$f$的依赖性得到改进。数值实验验证了理论结果，并表明高阶CIP-FEM能显著降低污染误差。